|
CoCoA 4.2 Online Help |
Online Help by David Perkinson
This document is a translation of the CoCoA online help system into HTML format. Please report errors to cocoa@dima.unige.it.
- Table of Contents for Online Manual
- List of Commands
Online Manual: Table of Contents
- Part 0.
- Part 1. Introduction to CoCoA
- Part 2. The CoCoA Programming Language
- Part 3. Doing Mathematics with CoCoA
- Part 4. Working the System
ONLINE MANUAL
0.1 Preamble
0.1.1 Version
------------------------------------------------------- --- ___/ ___/ \ --- -- / _ \ / _ \ , \ -- -- \ | | \ | | ___ \ -- --- ____, __/ ____, __/ _/ _\ --- ------------------------------------------------------- -- Online Help : CoCoA 4.2 -- -- Author : David Perkinson -- -- email : davidp@reed.edu -- -- addr : Reed College, Portland, OR, USA -- -- date : 02 August 2002 -- -------------------------------------------------------
0.1.2 Preface
CoCoA is a system for doing "Computations in Commutative Algebra." It is one of the projects of an active research team in Computer Algebra at the University of Genova, Italy, whose current members are: Lorenzo Robbiano (team manager), John Abbott, Anna Bigatti, Massimo Caboara, Martin Kreuzer, David Perkinson, and occasionally other researchers and students. Notable contributions from outside the team have been received from Volker Augustin and Arndt Wills. Much of the original code for CoCoA was written by Gianfranco Niesi and Antonio Capani.
0.1.3 System Distribution
CoCoA is public domain software, available at
http://cocoa.dima.unige.it
where there is also a great deal of information about the system.
Besides the main site in Genoa, Italy, there are two mirrors of the
official distribution:
Germany:
http://www.physik.uni-regensburg.de/~krm03530/cocoa/distribute.html
USA, West Coast:
http://www.reed.edu/cocoa
CoCoA is distributed freely under the following condition: any
research activity which uses CoCoA should quote the authors in the
following form:
A. Capani, G. Niesi, L. Robbiano,
CoCoA, a system for doing Computations in Commutative Algebra,
Available via anonymous ftp from: cocoa.dima.unige.it
The system can be freely redistributed to other users. New users
should notify the CoCoA team at the email address below so they can be
included in a users list. All members of this list will be kept up to
date about the progress of the system.
The system is distributed "as is." The authors make no guarantee
about the fitness of the system for any particular application. They
will not be liable for any direct, indirect, special, incidental or
consequential damages in connection with the use of the system or of
the manual.
Bug reports, questions, and suggestions should be sent to the
following email address:
cocoa@dima.unige.it
Comments on the online help system would be greatly appreciated. They
may be sent to the above address or to davidp@reed.edu.
0.1.4 System Requirements
CoCoA 4.2 runs on most common platforms. Please visit the CoCoA web site to see whether it is available for the platform you wish to use. If the version you seek is absent let us know, and we will try to rectify the situation.
0.1.5 Copyright and Trademarks
The product names mentioned in this manual are trademarks or registered trademarks of their manufacturers.
0.1.6 Acknowledgments
The CoCoA project is partly supported by: * Department of Mathematics, University of Genova * Department of Computer Science, University of Genova * Consiglio Nazionale delle Ricerche * Ministero dell'Universita` e della Ricerca Scientifica e Tecnologica The author of the online help package would like to thank Tony Geramita, Lorenzo Robbiano, and the CoCoA team. Thanks to Antonio Capani and Gianfranco Niesi for patiently answering my questions and for allowing me to use their "CoCoA User's Manual, version 3.0b Draft," from which the online help package borrows, sometimes verbatim.
Introduction to CoCoA
1.1 The CoCoA System
- An Overview of the System
- System Structure
- Contributions
- CoCoA and Macaulay
- Pointers to the Literature
1.1.1 An Overview of the System
CoCoA is a computer algebra system for doing "Computations in
Commutative Algebra." Since its first version CoCoA has been designed
to offer maximum ease of use and flexibility to the mathematician with
little or no knowledge of computers. It is able to perform simple and
sophisticated operations on multivariate polynomial rings and on
various data connected with them (ideals, modules, matrices, rational
functions).
The system is capable of performing basic operations such as
* sums, products, powers, derivatives, gcd, lcm of polynomials;
* sums, products, powers, derivatives of rational functions;
* sums, products, powers of ideals;
* sums of modules;
* sums, products, powers, determinants, adjoints of matrices;
* operations between heterogeneous values, like the product
between an ideal and a polynomial, etc.
Besides these, more advanced operations are available. For example:
* Groebner bases of ideals and modules;
* syzygies of ideals and modules;
* minimal free resolutions of ideals and modules;
* intersection and division of ideals and modules;
* inclusion and equality test for ideals and modules;
* elimination of indeterminates;
* homogenization of ideals;
* Poincare series and Hilbert functions;
* factorization of polynomials;
* saturation of toric ideals.
Every computation is performed within a "current" or "active"
ring, but the user can easily define and switch between many rings in
a single CoCoA session.
CoCoA includes an extensive Pascal-like programming language, CoCoAL,
that allows the user to customize the system and extend the embedded
library.
1.1.2 System Structure
CoCoA is written in C language and is available on several platforms. The core of the system is an implementation of Buchberger's algorithm for computing Groebner bases of ideals and modules over a polynomial ring whose coefficient ring is a field, and a variation of it for computing syzygies. The algorithm has been optimized in several directions and is used as a building block for many operations. Most users can, however, completely ignore the theory of Groebner bases and even their existence: CoCoA will do all the necessary "Groebner stuff" in the background. However, for optimum use of the system some knowledge of the theory is helpful. The system interacts with the external world using a Low Level Protocol (LLP) that is independent of any machine architecture. High level inputs from the user are translated into LLP-requests to the CoCoA kernel. The LLP-answers are then translated back into high level, user-readable outputs.
1.1.3 Contributions
Important parts of the system have been developed by: John Abbott: factorization and linear algebra Anna Bigatti: Hilbert-Poincare series, toric ideals Massimo Caboara: advanced ideal and module operations Martin Kreuzer: testing and debugging David Perkinson: online help Volker Augustin: graphical user interface Arndt Wills: graphical help system
1.1.4 CoCoA and Macaulay
Most of the standard scripts from the computer algebra system
Macaulay (Classic) have been translated into CoCoAL. For
information, see the CoCoA homepage at
http:://cocoa.dima.unige.it
mirrors
Germany:
http://www.physik.uni-regensburg.de/~krm03530/cocoa/distribute.html
USA, West Coast:
http://www.reed.edu/cocoa
1.1.5 Pointers to the Literature
The following are articles which may be of interest to CoCoA users. Many of the algorithms discussed in these articles have been implemented in CoCoA. J. Abbott, "Univariate factorization over the integers," Preprint (1998). A.M. Bigatti, "Computations of Hilbert-Poincare Series," J. Pure Appl. Algebra, 119/3, 237--253 (1997). A.M. Bigatti, A. Capani, G. Niesi, L. Robbiano, "Hilbert-Poincare Series and Elimination Problems," Preprint (1998). A.M. Bigatti, R. La Scala, L. Robbiano, "Computing Toric Ideals," Preprint (1998). A.M. Bigatti, L. Robbiano, "Borel Sets and Sectional Matrices," Annals of Combinatorics, 1, 197--213, (1997) M. Caboara, P. Conti, and C. Traverso, "Yet Another Ideal Decompostion Algorithm," AAECC-12, Springer LNCS 1255, 39--54, (1997). M. Caboara, G. De Dominicis, L. Robbiano, "Multigraded Hilbert Functions and Buchberger Algorithm," In Proc. ISSAC '96, 72--78 (1996), Y.N. Lakshman, editor, New York. ACM Press. A. Capani, G. De Dominicis, "Web Algebra," In Proc. of WebNet 96. Association for the Advancement of Computing in Education (AACE) Charlottesville, USA, (1996). A. Capani, G. De Dominicis, G. Niesi, L. Robbiano, "Computing Minimal Finite Free Resolutions," J. Pure Appl. Algebra, 117/118, 105--117, (1997). A. Capani, G. Niesi, "The CoCoA 3 Framework for a Family of Buchberger-like Algorithms," In Groebner Bases and Applications (Proc. of the Conf. 33 Years of Groebner Bases) , London Math. Soc. Lecture Notes Series, Vol. 251, B. Buchberger and F. Winkler eds., Cambridge University Press, p. 338--350, (1998). A. Capani, G. Niesi, "CoCoA 3.0 User's Manual," (1995). A. Capani, G. Niesi, L. Robbiano, "Some Features of CoCoA 3," Comput. Sci. J. of Moldova 4, 296--314, (1996). A. Giovini, T. Mora, G. Niesi, L. Robbiano, C. Traverso, "`One sugar cube, please' or selection strategies in the Buchberger algorithm," In Proc. ISSAC '91, 49--54, Stephen M. Watt, editor, New York, ACM Press, (1991). A. Giovini and G. Niesi, "CoCoA: a user-friendly system for commutative algebra," In Design and Implementation of Symbolic Computation Systems -- International Symposium DISCO'90, Lecture Notes in Comput. Sci., 429, 20--29, Berlin, Springer Verlag, (1990). B. Sturmfels, "Groebner Bases and Convex Polytopes", AMS University Lecture Series, Vol. 8 (1995). SOME BOOKS AND ARTICLES MENTIONING CoCoA: W. W. Adams, P. Loustaunau, "An Introduction to Groebner Bases," Graduate Studies in Mathematics, AMS, Providence, R.I. (1994). D. Cox, J. Little, D. O'Shea, "Ideals, Varieties, and Algorithms," Springer-Verlag, New York (1992). M. Kreuzer, L. Robbiano, "Computational Commutative Algebra 1" Springer-Verlag, (2000). L. Robbiano, "Groebner Bases and Statistics," in "Groebner Bases and Applications," (Proceedings of the Conference: 33 Years of Groebner Bases), LMS Lecture Note Series, Vol. 251, B. Buchberger and F. Winkler eds., Cambridge University Press, p. 79--204 (1998).
1.2 Tutorial
- A Tutorial Introduction to CoCoA
- Setting Up CoCoA for the Tutorial
- Entering Commands
- Examples of Entering Commands
- More on Entering Commands
- After the Tutorial
- Arithmetic
- Variables
- The Variable 'It'
- Making Lists
- Setting Up a Ring
- A Groebner Basis Example
- Eliminating Variables
- Using More Than One Ring
- Substitutions
- First Functions
- More First Functions
- Rings Inside User-Defined Functions
- Rational Normal Curve
- Generic Minors
- Leading Term (Initial) Ideals, Generic Polynomials
- Ring Mapping Example
- Output to a File
- Finite Point Sets: Buchberger-Moeller
- Syzygies and Resolution Example
- Factoring Polynomials
1.2.1 A Tutorial Introduction to CoCoA
This is an introduction to CoCoA, mainly through examples. It is just a small part of the online CoCoA manual. To learn more about finding information in the manual, enter '?' (without the quotes). For now, the only essential command to know is 'H.Browse()'. Entering 'H.Browse();' or 'H.Browse(1);' will display the next section of the manual. 'H.Browse(0);' will redisplay the current section, and 'H.Browse(-1);' will display the previous section. Start browsing the tutorial now by entering 'H.Browse();'.
1.2.2 Setting Up CoCoA for the Tutorial
Some of the information from the online manual (and results of CoCoA calculations, in general) may scroll off of the screen. It is best to run CoCoA from a system that has scrollable windows with enough room to hold the output from each CoCoA command. Under X-windows, you might try running CoCoA from an xterm started with the command 'xterm -sb -sl 512' (scroll bar enabled, saving 512 lines). In addition, you may want to increase the vertical size of your window, e.g., 'xterm -sb -sl 512 -geometry 80x40' under X-windows. Better yet: run CoCoA from a shell within Emacs. IF YOU DO NOT HAVE SCROLLABLE WINDOWS: enter the command 'H.SetMore();' to receive output 20 lines at a time. As long as there is output waiting to be printed, you will be prompted to enter 'More();' to get the next 20 lines. You may type 'H.SetMore(N)' in order to get N lines at a time instead of 20. Enter 'H.Browse();' to continue. (The 'Browse' command will be assumed from now on.)
1.2.3 Entering Commands
While reading the tutorial it is highly recommended that you have a copy of CoCoA running in a separate window in order to play with the examples presented. To execute a CoCoA command, type it into the window, ending it with a semicolon, then press the 'return' key or 'enter' key (depending on your system). A command can run over several lines; CoCoA will wait for a semicolon before processing the command. Also, several commands may be written on a single line (each ending with a semicolon). See the next section for examples. IMPORTANT NOTES for Macintosh and Windows users: 1. If your system has two 'enter' keys or has an 'enter' key and a separate 'return' key, it may be that only one of these keys will function to enter CoCoA commands. Experiment. (On some systems the 'enter' key on the far right of the keyboard is the one to use.) 2. To enter multiple lines on Macintosh and Windows systems, one needs to highlight the lines using the mouse before hitting the 'enter' key. Otherwise, the enter key just feeds the line containing the cursor to CoCoA.
1.2.4 Examples of Entering Commands
Here are some examples of entering simple commands in CoCoA. (Macintosh and Windows users please remember the special instructions made in the previous section.) >EXAMPLE< ------------------------------- 1+1; -- a simple command 2 ------------------------------- 1 + -- the same command spread over several lines 1; 2 ------------------------------- 1+1; 2+2; 3+3; -- several commands on a single line 2 ------------------------------- 4 ------------------------------- 6 ------------------------------- NOTE: The output above appears just as it does in my CoCoA window. The examples in the online manual will often be annotated. When CoCoA encounters *double dashes*, as above, it regards the rest of the line as a comment.
1.2.5 More on Entering Commands
A common way to enter CoCoA commands is to first type them into a text editor, then cut-and-paste into the CoCoA window. For one, this saves a lot of retyping in the case of syntax errors. In addition, one may save a sequence of commands in a file and read them into a CoCoA session with the 'Source' command (or equivalently, with '<<'). For instance, if a sequence of CoCoA commands is saved in the file 'MySession', one may enter << 'MySession'; to run the commands (give the full pathname, or store your file in the cocoa directory). User-defined functions are often stored in files and sourced in this way.
1.2.6 After the Tutorial
Browsing through the examples in this tutorial may enable you to
solve your particular problems. If not, or to learn more about CoCoA
in general, enter '?' to learn more about CoCoA's online help. For
example, you will see that 'H.Command("")' gives a long annotated
list of CoCoA commands, and '?keyword' will look for information about
"keyword" in the manual.
With the following section, the tutorial proper begins. To see a
table of contents for the tutorial, enter 'H.Toc(1,2)'. The command
'H.Browse()' will then continue with the tutorial, or you may give the
title of a tutorial section to '?' to skip to that section.
1.2.7 Arithmetic
Here are some first examples; they illustrate CoCoA evaluating arithmetic expressions. >EXAMPLE< (2+3)(1+1); -- multiplication, as usual 10 ------------------------------- 2^10; 1024 ------------------------------- 2+2/3; 8/3 ------------------------------- 1.5+2.3; -- decimals are converted to fractions 19/5 ------------------------------- Mod(27,5); 2 ------------------------------- 2*3; 6 ------------------------------- Fact(4); 24 ------------------------------- For multiplication, one may use '*', parentheses, or just a space.
1.2.8 Variables
Results from CoCoA calculations can be stored in variables. Variables, like CoCoA functions, must begin with a capital letter or an error will result. >EXAMPLE< A := 3; 2A; 6 ------------------------------- b := 7; ERROR: parse error in line 12 of device stdin ------------------------------- 7 ------------------------------- B := 7; A^2 + B; 16 -------------------------------
1.2.9 The Variable 'It'
When CoCoA evaluates an expression, the result is usually assigned to the special CoCoA variable named 'It'. >EXAMPLE< 1+1; 2 ------------------------------- It; 2 ------------------------------- It+1; 3 ------------------------------- It; 3 ------------------------------- X := 17; -- 'It' is not changed if there is no output It; 3 ------------------------------- X+It; 20 -------------------------------
1.2.10 Making Lists
The following example illustrates the use of lists in CoCoA.
>EXAMPLE<
L := [2,3,"a string",[5,7],3,3]; -- L is now a list
L[3]; -- here is the 3rd component of L
a string
-------------------------------
L[4]; -- the 4th component of L is a list, itself
[5, 7]
-------------------------------
L[4][2]; -- the 2nd component of the 4th component of L
7
-------------------------------
L[4,2]; -- same as above
7
-------------------------------
Append(L,"new");
L;
[2, 3, "a string", [5, 7], 3, 3, "new"]
-------------------------------
-- insert 8 as the 4th component of L, shifting the other
-- entries to the right:
Insert(L,4,8);
L;
[2, 3, "a string", 8, [5, 7], 3, 3, "new"]
-------------------------------
Remove(L,4); -- remove it again
L;
[2, 3, "a string", [5, 7], 3, 3, "new"]
-------------------------------
Len(L); -- the number of components of L
7
-------------------------------
Set(L); -- same as L but with repeats removed
[2, 3, "a string", [5, 7], "new"]
-------------------------------
1..5; -- a range of values
[1, 2, 3, 4, 5]
-------------------------------
[ X^2 | X In 1..5]; -- a useful way to make lists
[1, 4, 9, 16, 25]
-------------------------------
[1,2] >< [3,4] >< [5]; -- Cartesian product: use a greater-than
-- sign '>' and a less-than sign '<' to make
-- the operator '><'.
[[1, 3, 5], [1, 4, 5], [2, 3, 5], [2, 4, 5]]
-------------------------------
1.2.11 Setting Up a Ring
A CoCoA session automatically starts with the default ring, R = Q[t,x,y,z]. The command 'Use' is used to change rings. The following example shows how to create the ring Z/(5)[a,b,c] (the coefficient ring is the integers mod 5). Once the ring has been declared, one may start to play with polynomials, ideals, modules, and other constructions in that ring. In the ring declaration, the indeterminates are optionally separated by commas, and note the use of two colons. Some details on handling several rings is provided below in the section of the Tutorial entitled "Using More Than One Ring." >EXAMPLE< Use S ::= Z/(5)[a,b,c]; F := a-b; I := Ideal(F^2,c); I; Ideal(a^2 - 2ab + b^2, c) ------------------------------- J := Ideal(a-b); I + J; Ideal(a^2 - 2ab + b^2, c, a - b) ------------------------------- Minimalized(It); -- find a minimal set of generators for I+J Ideal(a - b, c) -------------------------------
1.2.12 A Groebner Basis Example
A Groebner basis of an ideal I is calculated with the command 'GBasis(I)', as illustrated in the following example. Let r be a root of the equation x^7-x-1 over the rationals. The minimal polynomial of (4r-1)/r^3 can be found by computing the reduced Groebner basis of the ideal (x^7-x-1,x^3y-4x+1) with respect to the lexicographic term-ordering with x>y. >EXAMPLE< Use R ::= Q[x,y], Lex; Set Indentation; -- to improve the appearance of the output G := GBasis(Ideal(x^7-x-1,x^3y-4x+1)); G; [ 1602818757152090759440/34524608236181199361x - 4457540/5875764481y^7 - 47746460716124220/34524608236181199361y^6 + 890175715271333840/34524608236181199361y^5 - 3541992534667352220/34524608236181199361y^4 - 55943894513139464160/34524608236181199361y^3 - 56473654361333280980/34524608236181199361y^2 - 27971979712025453040/34524608236181199361y - 400704689288022689860/34524608236181199361, 1/16384y^7 - 5/16384y^6 + 147/16384y^4 + 5/128y^3 - 31/16384y^2 + 17/128y - 20479/16384] ------------------------------- Len(G); 2 ------------------------------- F := 16384*G[2]; -- clear denominators F; y^7 - 5y^6 + 147y^4 + 640y^3 - 31y^2 + 2176y - 20479 ------------------------------- The Groebner basis is reported as a list with two elements. The second gives a univariate polynomial which is the minimal polynomial for r. Note that the statement declaring the ring includes the modifier, 'Lex'. Without this modifier, the default term-ordering, DegRevLex, is used. The command 'Set Indentation' forces each polynomial of the Groebner basis to be printed on a new line.
1.2.13 Eliminating Variables
The Cartesian equations of the space curve parametrized by
t --> (t^31+t^6,t^8,t^10)
can be found by eliminating the indeterminate t in the ideal
(t^31+t^6-x, t^8-y, t^10-z).
>EXAMPLE<
Use R ::= Q[t,x,y,z];
Set Indentation;
Elim(t,Ideal(t^31+t^6-x, t^8-y, t^10-z));
Ideal(
y^5 - z^4,
-y^4z^5 + y^4 - 2xy^2z + x^2z^2,
-z^8 - 2xy^3 + x^2yz + z^3,
2xy^4z^4 + yz^7 + 3x^2y^2 - 2x^3z - yz^2,
-y^2z^6 - 1/2xz^7 + 1/2x^3y + y^2z - 3/2xz^2,
-1/3x^2y^4z^3 - y^3z^5 - 2/3xyz^6 + 1/3x^4 + y^3 - 4/3xyz)
-------------------------------
With the command 'Elim', CoCoA automatically switches to a
term-ordering suitable for eliminating the variable t, then changes
back to the declared term-ordering (in this case the default
term-ordering, DegRevLex).
One may see the entire Groebner basis for our ideal with respect to
the elimination term-ordering for t as follows:
>EXAMPLE<
Use R ::= Q[t,x,y,z], Elim(t);
Set Indentation;
GBasis(Ideal(t^31+t^6-x, t^8-y, t^10-z));
[
-t^2y + z,
y^5 - z^4,
-t^6 - tz^3 + x,
-tz^4 - y^2 + xz,
-ty^2 + txz - y^4z,
-y^4z^5 + y^4 - 2xy^2z + x^2z^2,
-z^8 - 2xy^3 + x^2yz + z^3,
2xy^4z^4 + yz^7 + 3x^2y^2 - 2x^3z - yz^2,
tx^2y - tz^2 - y^2z^3 - xz^4,
2txyz^3 - z^7 - x^2y + z^2,
-y^2z^6 - 1/2xz^7 + 1/2x^3y + y^2z - 3/2xz^2,
t^2x - tx^2z^2 + xy^4z^2 + yz^5 - y,
t^2z + 2txz^3 - y^4z^3 - x^2,
-3tx^2z^3 + 2xy^4z^3 + yz^6 + x^3 - yz,
-1/3x^2y^4z^3 - y^3z^5 - 2/3xyz^6 + 1/3x^4 + y^3 - 4/3xyz,
1/3tx^3 - 1/3tyz - 1/3x^2y^4 - 1/3y^3z^2 - 1/3xyz^3]
-------------------------------
1.2.14 Using More Than One Ring
In CoCoA, every calculation takes place in a "current" or
"active" ring. Ring-dependent objects defined by the user such as
polynomials, ideals, and modules are automatically labeled by the
current ring. Objects that do not depend essentially on the ring,
e.g., lists or matrices of integers, do not get labeled.
CoCoA automatically starts with the ring R = Q[x,y,z]. The
following example illustrates setting up a ring with the construction
'::=' and changing rings with the command 'Use'. One may temporarily
change rings with the command 'Using'. The example assumes that you
do not already have a ring with identifier "S".
>EXAMPLE<
Use R ::= Q[x,y]; -- declare and use a ring R
F := (x+y)^3;
F;
x^3 + 3x^2y + 3xy^2 + y^3
-------------------------------
M := [1,'test',2];
S ::= Q[x,y,z,a,b]; -- declare a ring S with indeterminates x,y,z,a,b
Use S; -- switch to the ring S
F; -- F is labeled by ring R
R :: x^3 + 3x^2y + 3xy^2 + y^3
-------------------------------
M; -- this list is not labeled R since it's elements are not
-- ring dependent (e.g., '1' is considered a separate integer, not
-- part of the ring R)
[1, "test", 2]
-------------------------------
F := Ideal(a^2+b^2); -- change the definition of F
Use R; -- switch back to R
F; -- the old F no longer exists
S :: Ideal(a^2 + b^2)
-------------------------------
GBasis(F); -- built in functions automatically recognize the ring
[S :: a^2 + b^2]
-------------------------------
1.2.15 Substitutions
To substitute a list of numbers or polynomials for the indeterminates (in the order specified by the definition of the ring), one may use the function 'Eval'. To substitute out of order, use the function 'Subst'. >EXAMPLE< Use R ::= Q[x,y,z]; F := x^2+y^2+z^2; Eval(F,[1]); -- substitute x=1 y^2 + z^2 + 1 ------------------------------- Eval(F,[1,2,3]); -- substitute x=1, y=2, z=3 14 ------------------------------- Subst(F,y,2); -- substitute y=2 x^2 + z^2 + 4 ------------------------------- Eval(F,[x,2,z]); -- same as above x^2 + z^2 + 4 ------------------------------- Subst(F,[[y,y^2],[z,z^2]]); -- substitute y^2 for y, z^2 for z y^4 + z^4 + x^2 ------------------------------- Eval(Ideal(F),[x^2,z]); -- substitute x^2 for x, z for y Ideal(x^4 + 2z^2) -------------------------------
1.2.16 First Functions
CoCoA's library of functions can be easily extended with user-defined
functions. Longer functions are usually cut-and-pasted from a text
editor into a CoCoA session. If the functions are to be used
repeatedly, they can be saved in a separate text file and read into a
CoCoA session with the 'Source' command (or '<<'). The usual way to
define a function is with the syntax:
Define < FunctionName >(< argument list >) < Commands > EndDefine;
NOTE: Variables defined within a function are usually local to that
function and disappear after the function returns. Normally, the only
variables accessible within a function are the function's arguments
and local variables. (For the exceptions, see the section of the
manual entitled "Global Memory.")
>EXAMPLE<
Define Square(X) -- a simple function
Return X^2;
EndDefine;
Square(3);
9
-------------------------------
Define IsPrime(X) -- a more complicated function
If Type(X) <> INT Then Return Error('Expected INT') EndIf;
I := 2;
While I^2 <= X Do
If Mod(X,I) = 0 Then Return False EndIf;
I := I+1;
EndWhile;
Return TRUE;
EndDefine; -- end of function definition
IsPrime(4);
FALSE
-------------------------------
Define Test(A,B) -- a function with two arguments
Sum := A+B;
Prod := A*B;
PrintLn('The sum of ',A,' and ',B,' is ',Sum,'.');
Print('The product of ',A,' and ',B,' is ',Prod,'.');
EndDefine;
Test(3,5);
The sum of 3 and 5 is 8.
The product of 3 and 5 is 15.
-------------------------------
1.2.17 More First Functions
A user-defined function can have any number of parameters of any
type, even a variable number of parameters. Note that even a function
with no parameters must be called with parentheses.
>EXAMPLE<
Define Test1()
PrintLn('This is a function with no parameters.');
For I := 1 To 10 Do
Print(I^2,' ');
EndFor;
EndDefine;
Test1();
This is a function with no parameters.
1 4 9 16 25 36 49 64 81 100
-------------------------------
Define Test2(...) -- a variable number of parameters
If Len(ARGV) = 0 Then -- parameters are stored in the list ARGV
Return 'Wrong number of parameters';
Elsif Len(ARGV) = 1 Then
Print('There is 1 parameter: ',ARGV[1]);
Else
Print('There are ',Len(ARGV),' parameters: ');
Foreach P In ARGV Do
Print(P,' ');
EndForeach;
EndIf;
EndDefine;
Test2(1,2,'string',3);
There are 4 parameters: 1 2 string 3
-------------------------------
1.2.18 Rings Inside User-Defined Functions
As mentioned earlier, user-defined functions cannot reference
(non-global) variables except those defined within the function or
passed as arguments. However, functions can refer to rings via their
identifiers and use them as one would outside of a function.
When a function is called, it assumes the current ring and performs
operations in that ring. One may define new rings which will exist
after the function returns, but one may not change the current ring
with the command 'Use'. However, one may *temporarily* use a ring
with the command 'Using'.
To make functions more portable, it may be useful to refer to the
current ring not by its name but by using the command 'CurrentRing'.
>EXAMPLE<
-- Example I.
-- Test uses the existing rings, R, S, and creates a new ring T.
-- While a (non-global) *variable* defined in a function will
-- automatically disappear, a ring (and its name) will not.
Use R ::= Q[x,y,z];
S ::= Q[a,b];
Define Test()
PrintLn (x+y)^2;
PrintLn S :: (a+b)^3;
T ::= Z/(5)[t];
I := T :: Ideal(t^2);
Print I;
EndDefine;
Test();
x^2 + 2xy + y^2
S :: a^3 + 3a^2b + 3ab^2 + b^3
T :: Ideal(t^2)
-------------------------------
I; -- the variable I was local to the function
ERROR: Undefined variable I
CONTEXT: I
-------------------------------
T; -- The function created the ring T. (Note: T is not a variable.)
Z/(5)[t]
-------------------------------
-- Example II.
-- The use of 'CurrentRing' within a function.
Define Poincare2(I)
Return Poincare(CurrentRing()/I);
EndDefine;
Use R ::= Q[x,y];
Poincare2(Ideal(x^2,y^2));
(1 + 2x + x^2)
-------------------------------
-- Example III.
-- Creating a ring with a user-supplied name. For more information,
-- see 'Var'.
Define Create(Var(R));
Var(R) ::= Q[a,b];
EndDefine;
Create('K');
K;
Q[a,b]
-------------------------------
Create('myring');
Var('myring');
Q[a,b]
-------------------------------
Use Var('myring'); -- make myring current
-- Example IV.
-- A more complicated example, creating rings whose names are
-- automatically generated. See 'NewId' and 'Var' for more
-- information.
Define CreateRing(I)
NewRingName := NewId();
Var(NewRingName) ::= Q[x[1..I]],Lex;
Return NewRingName;
EndDefine;
Use R ::= Q[x,y],DegRevLex;
Use S ::= Q[x,y,z],Lex;
N := 5;
For I := 1 To N Do
RingName := CreateRing(I); -- RingName is a string
Using Var(RingName) Do
PrintLn Indets();
EndUsing;
-- Destroy Var(RingName); -- uncomment if you want to destroy the tmp
-- ring
EndFor;
RingEnvs();
[x[1]]
[x[1], x[2]]
[x[1], x[2], x[3]]
[x[1], x[2], x[3], x[4]]
[x[1], x[2], x[3], x[4], x[5]]
-------------------------------
["Q", "Qt", "R", "S", "V#1", "V#3", "V#5", "V#7", "V#9", "Z"]
-------------------------------
1.2.19 Rational Normal Curve
In this example, we compute the ideal of the rational normal curve of
degree N in P^N then compute its Poincare series for a range of values
of N.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
Define Rational_Normal_Curve_Ideal(N)
-- first define the 2xN matrix whose 2x2 minors generate the ideal
M := NewMat(2,N);
For C := 0 To N-1 Do
M[1,C+1] := x[C];
M[2,C+1] := x[C+1]
EndFor;
-- then construct the generators of the ideal
L := [];
For C1 := 1 To N-1 Do
For C2 := C1+1 To N Do
P := M[1,C1] M[2,C2] - M[2,C1] M[1,C2];
-- determinant for columns C1,C2
Append(L,P)
EndFor
EndFor;
Return Ideal(L)
EndDefine;
For N := 3 To 5 Do
S ::= Q[x[0..N]],Lex;
PrintLn NewLine, 'degree ', N;
Using S Do -- switch, temporarily, to ring S
I := Rational_Normal_Curve_Ideal(N);
Print('Poincare series: ');
Poincare(S/I);
EndUsing;
PrintLn;
EndFor; -- for statement
degree 3
Poincare series: (1 + 2x[0]) / (1-x[0])^2
degree 4
Poincare series: (1 + 3x[0]) / (1-x[0])^2
degree 5
Poincare series: (1 + 4x[0]) / (1-x[0])^2
-------------------------------
1.2.20 Generic Minors
The following example computes the relations among the 2x2 minors of
a generic 2xN matrix for a range of values of N. Note the use of
indeterminates with multiple indices.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
Define Det_SubAlgebra(N)
L := [];
For C1 := 1 To N-1 Do
For C2 := C1+1 To N Do
P := y[C1,C2]-(x[1,C1] x[2,C2] - x[2,C1] x[1,C2]);
Append(L,P)
EndFor
Endfor;
Return Ideal(L)
EndDefine;
Define Det_SubAlgebra_Print(N) -- calculate and print relations
J := Det_SubAlgebra(N);
PrintLn NewLine,'N = ',N;
PrintLn 'Sub-algebra equations:';
PrintLn Gens(Elim(x,J))
EndDefine;
Set Indentation;
For N := 3 To 5 Do
S ::= Z/(32003)[y[1..(N-1),2..N],x[1..2,1..N]];
Using S Do
Det_SubAlgebra_Print(N);
EndUsing
EndFor;
N = 3
Sub-algebra equations:
[
0]
N = 4
Sub-algebra equations:
[
2y[1,4]y[2,3] - 2y[1,3]y[2,4] + 2y[1,2]y[3,4]]
N = 5
Sub-algebra equations:
[
2y[2,5]y[3,4] - 2y[2,4]y[3,5] + 2y[2,3]y[4,5],
2y[1,5]y[3,4] - 2y[1,4]y[3,5] + 2y[1,3]y[4,5],
2y[1,5]y[2,4] - 2y[1,4]y[2,5] + 2y[1,2]y[4,5],
2y[1,5]y[2,3] - 2y[1,3]y[2,5] + 2y[1,2]y[3,5],
2y[1,4]y[2,3] - 2y[1,3]y[2,4] + 2y[1,2]y[3,4]]
-------------------------------
1.2.21 Leading Term (Initial) Ideals, Generic Polynomials
The following example produces the leading term (initial) ideal of the ideal generated by three "generic" polynomials of degree 2 with respect to the lexicographic term-ordering. >EXAMPLE< Use R ::= Z/(32003)[x[1..4]],Lex; F := DensePoly(2); -- sum of all power-products of degree 2 L := [ Randomized(F) | I In 1..3 ]; -- randomize coefficients LT(Ideal(L)); Ideal(x[1]^2, x[1]x[2], x[1]x[3], x[2]^3, x[1]x[4]^2, x[2]^2x[3], x[2]^2x[4]^2, x[2]x[3]^3, x[2]x[3]^2x[4]^2, x[2]x[3]x[4]^4, x[2]x[4]^6, x[3]^8) -------------------------------
1.2.22 Ring Mapping Example
If R is the current ring and E is an object in another ring, then the
function 'Image' may be used to map E into R by substituting
polynomials from R for the indeterminates in E. (Also: see the
command 'BringIn' for a shortcut in certain cases.)
>EXAMPLE<
Use S ::= Q[a,b,c];
I := Ideal(a^2+b^2,ab-c^2);
Use R ::= Q[x,y]; -- the current ring is R
F := RMap(x+y,x-y,y^2); -- define a map F:S --> R sending a to x+y,
-- b to x-y, and c to y^2
Image(I,F); -- the image of I under F
Ideal(2x^2 + 2y^2, -y^4 + x^2 - y^2)
-------------------------------
1.2.23 Output to a File
The following example illustrates one way of saving CoCoA output to a
file.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
I := Ideal(t^2-x,t^5-y,t^7-z);
G := GBasis(I);
G;
[t^2 - x, -tx^2 + y, -x^3 + ty, -xy + z, -ty^2 + x^2z, txz - y^2, y^3 - tz^2]
-------------------------------
D := OpenOFile('MyFile'); -- open 'MyFile' for output from CoCoA
Print G On D; -- G is appended to the file 'MyFile'
Close(D);
Text can be read from files using 'OpenIFile' and 'Get', and commands
can be executed from files using 'Source' (or '<<'). One may also
keep a log of a CoCoA session (see 'OpenLog')
1.2.24 Finite Point Sets: Buchberger-Moeller
CoCoA includes an implementation of the Buchberger-Moeller algorithm for efficient computations dealing with finite sets of points. These functions include: * GBM, HGBM -- intersection of ideals for zero-dimensional schemes * IdealAndSeparatorsOfPoints -- ideal & separators for affine points * IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points * IdealOfPoints -- ideal of a set of affine points * IdealOfProjectivePoints -- ideal of a set of projective points * Interpolate -- interpolating polynomial * SeparatorsOfPoints -- separators for affine points * SeparatorsOfProjectivePoints -- separators for projective points Details about these functions may be found individually in the online manual. Briefly, the functions above may be used to find the ideal of a set of points in affine or projective space along with a reduced Groebner basis and separators. Separators are polynomials that take the value 1 on one of the points and 0 on the remainder. The Buchberger-Moeller algorithm works *much* faster than the straightforward technique involving intersecting ideals for the individual points. >EXAMPLE< Use R ::= Q[t,x,y,z]; Pts := GenericPoints(20); -- 20 random points in projective 3-space X := IdealAndSeparatorsOfProjectivePoints(Pts); Len(Gens(X.Ideal)); -- number of generators in the ideal 17 ------------------------------- Hilbert(R/X.Ideal); H(0) = 1 H(1) = 4 H(2) = 10 H(t) = 20 for t >= 3 ------------------------------- F := X.Separators[1]; [Eval(F,P)| P In Pts]; [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ------------------------------- Res(R/X.Ideal); -- the resolution of the ideal 0 --> R^10(-6) --> R^24(-5) --> R^15(-4) --> R -------------------------------
1.2.25 Syzygies and Resolution Example
The following example, among other things, computes the resolution of
ideals of sets of points.
>EXAMPLE<
Use R ::= Q[x,y,z];
X1 := [[0,0,1],[1,0,1],[2,0,1],[2,1,1]]; -- 4 points in the projective
-- plane
X2 := [[0,0,1],[1,0,1],[0,1,1],[1,1,1]]; -- 4 more points
I1 := IdealOfProjectivePoints(X1);
I2 := IdealOfProjectivePoints(X2);
Hilbert(R/I1); -- the Hilbert function of X1
H(0) = 1
H(1) = 3
H(x) = 4 for t >= 2
-------------------------------
Hilbert(R/I2) = Hilbert(R/I1); -- The Hilbert functions for X1 and X2
-- are the same
TRUE
-------------------------------
Res(R/I1); -- but the resolutions ...
0 --> R(-3)(+)R(-4) --> R^2(-2)(+)R(-3) --> R
-------------------------------
Res(R/I2); -- are different.
0 --> R(-4) --> R^2(-2) --> R
-------------------------------
Describe Res(R/I1); -- more information about the resolution for X1
Mat[
[xy - 2yz, y^2 - yz, x^3 - 3x^2z + 2xz^2]
]
Mat[
[y - z, x^2 - xz],
[-x + 2z, 0],
[0, -y]
]
-------------------------------
Syz(I1,1); -- the first syzygy module for X1
Module([y - z, -x + 2z, 0], [x^2 - xz, 0, -y])
-------------------------------
1.2.26 Factoring Polynomials
CoCoA can factor a polynomial over its coefficient ring.
>EXAMPLE<
Use R ::= Q[x,y];
F := x^12 - 37x^11 + 608x^10 - 5852x^9 + 36642x^8 - 156786x^7 + 468752x^6
- 984128x^5 + 1437157x^4 - 1422337x^3 + 905880x^2 - 333900x + 54000;
Factor(F);
[[x - 2, 1], [x - 4, 1], [x - 6, 1], [x - 3, 2], [x - 5, 3], [x - 1, 4]]
-------------------------------
F := (x+y)^2*(x^2y+y^2x+3);
F;
x^4y + 3x^3y^2 + 3x^2y^3 + xy^4 + 3x^2 + 6xy + 3y^2
-------------------------------
Factor(F); -- multivariate factorization
[[x^2y + xy^2 + 3, 1], [x + y, 2]]
-------------------------------
Use Z/(37)[x];
Factor(x^6-1);
[[x - 1, 1], [x + 1, 1], [x + 10, 1], [x + 11, 1], [x - 11, 1], [x - 10, 1]]
-------------------------------
The CoCoA Programming Language
2.1 Introduction to CoCoA Programming
2.1.1 An Overview of CoCoA Programming
The CoCoA system includes a full-fledged high level programming language, CoCoAL, complete with loops, branching, scoping of variables, and input/output control. The language is used whenever one issues commands during a CoCoA session. A sequence of commands may be stored in a text file and then read into a CoCoA session using the 'Source' command (or '<<'). The most important construct in CoCoA programming is the user-defined function, created with 'Define'. A user-defined function can take any number of arguments, of any types, perform CoCoA commands, and return values. Collections of these functions can be stored in text files, as mentioned in the preceding paragraph, or formed into CoCoA "packages," to be made available for general use.
2.2 Language Elements
2.2.1 Character Set and Special Symbols
The CoCoA character set consists of the 26 lower case letters, the 26
upper case letters, the 10 digits and the special characters listed in
the table below. Note that the special character | looks a bit
different on some keyboards (its ascii code is 124).
------------------------------------------------------
| blank _ underscore ( left parenthesis |
| + plus = equal ) right parenthesis |
| - minus < less than [ left bracket |
| * asterisk < greater than [ right bracket |
| / slash | vertical bar ' single quote |
| : colon . period " double quote |
| ^ caret ; semicolon |
| , comma % percent |
------------------------------------------------------
Special Characters
The character-groups listed in the table below are special symbols in CoCoA
----------------------------------------------------------
| := assign .. range |
| << input from // start line comment |
| <> not equal -- start line comment |
| <= less than or equal to /* start comment |
| >= greater than or equal to */ end comment |
| >< Cartesian product :: ring casting |
| ::= ring definition ... dots |
----------------------------------------------------------
Special Character-groups
2.2.2 Identifiers
There are two types of identifiers or names.
* Identifiers of ring indeterminates. They must begin with lower case
letters.
* Predefined or user-defined names (functions and CoCoAL variables).
They must begin with upper case letters.
2.2.3 Names of Indeterminates
Each indeterminate of a polynomial ring may have one of the following
forms:
* a single lower case letter;
* a single lower case letter, indexed by one or more integer
expressions, separated by commas, and enclosed by square
brackets, e.g. x[1,3+5,2].
2.2.4 Reserved Names
The names in the following tables are reserved and cannot be used
otherwise. The names in the first table are case insensitive
(e.g. CLEAR, Clear and ClEaR are all reserved). The names in the
second table are case sensitive.
----------------------------------------------------
| Alias And Block Catch Ciao |
| Clear Cond Define Delete Describe |
| Destroy Do Elif Else End |
| EndBlock EndCatch EndCond EndDefine EndFor |
| EndForeach EndIf EndPackage EndRepeat EndUsing |
| EndWhile Eof False For Foreach |
| Global Help If In IsIn |
| NewLine Not On Or Package |
| Print PrintLn Quit Repeat Record |
| Return Set Skip Source Step |
| Then Time To True Unset |
| Until Use Using Var While |
----------------------------------------------------
Case insensitive reserved names
---------------------------------------------
| BOOL DegLex DegRevLex DEVICE ERROR |
| FUNCTION IDEAL INT LIST Lex |
| MAT MODULE NULL Null PANEL |
| POLY PosTo RAT RATFUN RING |
| STRING TAGGED ToPos TYPE VECTOR |
| Xel ZMOD |
--------------------------------------------
Case sensitive reserved names
2.2.5 Comments
A comment begins with the symbol /* and ends with the symbol */. Comments may contain any number of characters and are always treated as white space. Comments may be nested and may span several lines. In addition, text starting with // or -- up to the end of a line is also considered comment. >EXAMPLE< // This is a line comment Print 1+1; -- a command followed by a comment 2 ------------------------------- /* example of /* nested */ comment */
2.2.6 Data Types
Each CoCoA object has a type. The possible types are: BOOL : A boolean. The boolean constants are TRUE and FALSE. DEVICE : For input/output. ERROR : Objects of this type are used in error-handling. FUNCTION : A CoCoA-defined function. IDEAL : An ideal. INT : An arbitrary precision integer. LIST : A list. MAT : A matrix. MODULE : A submodule of a free module. NULL : The null constant is Null. POLY : A polynomial. RAT : An arbitrary precision rational number. RATFUN : A rational function. RECORD : A record is a set of bindings, name --> object. RING : A base ring, a polynomial ring, or a quotient ring. STRING : A string. TAGGED : Used to help in printing complicated objects. TYPE : Returned by the function 'Type', for example. VECTOR : A vector. ZMOD : An integer modulo another integer, e.g. 3 % 5. The types are partially ordered by inclusion of the sets that they represent, as follows: MAT < LIST VECTOR < LIST INT < RAT < POLY < RATFUN ZMOD < POLY < RATFUN (if compatible). In the manual, OBJECT is used to refer to an arbitrary CoCoA type. It is not a type itself.
2.2.7 Commands and Functions for Data Types
The following are commands and functions for data types:
* Cast -- type conversion
* Max, Min -- a maximum or minimum element of a sequence or list
* Shape -- extended list of types involved in an expression
* Type -- the data type of an expression
* TypeOfCoeffs -- type of the coefficients of the current ring
* Types -- lists all data types
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
2.3 Operators
- CoCoA Operators
- Algebraic Operators
- Relational Operators
- Boolean Operators
- Selection Operators
- Range Operator
2.3.1 CoCoA Operators
In CoCoA there are 5 main types of operators: algebraic
operators, relational operators, boolean operators, selection
operators, and the range operator. There is also an n-ary operator ><
for forming Cartesian products of lists and an operator :: used in
defining rings.
The meaning of an operator depends on the types of its operands; the +
in the expression A + B represents the sum of polynomials, or of
ideals, or of matrices, etc. according to the type of A and B.
The multiplication symbol '*' can always be omitted. The expression
F(E) is intrinsically ambiguous; it can be the variable F multiplied
by the parenthesized expression E, or the application of the function
F to the argument E. CoCoA always interprets this expression in the
latter way. In the former case the user must separate F from the left
parenthesis with a blank or an `*'.
The CoCoA operators are, from the highest to the lowest priority:
[] . (selection operators)
^ %
+ - (as unary operators)
* : /
+ - (as binary operators)
..
= <> < <= > >=
IsIn
Not
And
OR
Operations with equal priority are performed from left to right.
When in doubt, parentheses may be used to enforce a particular order of
evaluation.
Furthermore there is the n-ary operator >< (made by using a greater
than sign '>' and a less than sign '<') for making Cartesian
products of lists (see '><' or 'Cartesian Product') and the operator
:: for defining rings (see 'New Rings' and 'Use').
2.3.2 Algebraic Operators
The algebraic operators are:
+ - * / : ^
The following table shows which operations the system can perform
between two objects of the same or of different types; the first
column lists the type of the first operand and the first row lists the
type of the second operand. So, for example, the symbol `:' in the box
on the seventh row and fourth column means that it is possible to
divide an ideal by a polynomial.
INT RAT ZMOD POLY RATFUN VECTOR IDEAL MODULE MAT LIST
INT +-*/^ +-*/ * +-*/ +-*/ * * * * *
RAT +-*/^ +-*/ +-*/ +-*/ * * * * *
ZMOD *^ +-*/ +-*/ +-*/ * * * * *
POLY +-*/^ +-*/ +-*/ +-*/ +-*/ * * * * *
RATFUN +-*/^ +-*/ +-*/ +-*/ +-*/ * *
VECTOR * * * * +-
IDEAL *^ * * *: +*: *
MODULE * * * * : * +:
MAT *^ * * * * +-*
LIST * * * * * +-
Algebraic operators
Remarks:
* Let F and G be two polynomials. If F is a multiple of G, then
F/G is the polynomial obtained from the division of F by G,
otherwise F/G is a rational function (common factors are
simplified). The functions 'Div' and 'Mod' can be used to get the
quotient and the remainder of a polynomial division.
* Let L_1 and L_2 be two lists of the same length. Then L_1 + L_2 is
the list obtained by adding L_1 to L_2 componentwise.
* If I and J are both ideals or both modules, then I : J is the
ideal consisting of all polynomials f such that fg is in I for all
g in J. The division of an ideal I by a polynomial f is the
division of I by the ideal generated by f.
2.3.3 Relational Operators
The relational operators are:
= <> < > <= >= IsIn
The operator IsIn is quite flexible: see its manual for an explanation.
The other relational operators can be applied to two objects of the same type.
The admissible types for = and <> are:
NULL, BOOL, STRING, TYPE, INT, RAT, ZMOD, POLY, RATFUN,
VECTOR, IDEAL, MODULE, MAT, LIST.
The admissible types for the other relational operators are:
STRING, TYPE, INT, RAT, IDEAL, MODULE.
The meaning of < for string is the lexicographic comparison. For
ideals and modules A < B and A <= B both mean that A is (not
necessarily strictly) contained in B.
NOTE: It is sometimes hard to judge the type of an expression from the
appearance of CoCoA output, leading to confusing results from the
relational operators. Here is an simple example:
>EXAMPLE<
L:="3";
L;
3
-------------------------------
L=3;
FALSE
-------------------------------
Type(L);
STRING
-------------------------------
Type(3);
INT
-------------------------------
>END OF EXAMPLE<
Tagged expressions are especially prone to causing confusion of this
sort.
2.3.4 Boolean Operators
The boolean operators are:
Not And Or
see 'Introduction to Booleans'.
2.3.5 Selection Operators
The selection operators are
[] .
Let N be of type INT and let L be of type STRING, VECTOR, LIST, or
MAT. Then the meaning of L[N] depends on the type of L as explained
in the following table:
------------------------------------------------------------
| Type of L Meaning of L[N] |
------------------------------------------------------------
| STRING string consisting of the N-th character of L. |
| VECTOR N-th component of L |
| LIST N-th element of L |
| MAT N-th element of L |
------------------------------------------------------------
Selection Operator
If N is an identifier and L is of type RECORD, then L.N indicates the
object contained in the field N of the record L (see 'Introduction to
Records' and 'Records').
2.3.6 Range Operator
If M and N are of type INT, then the expression:
M .. N
returns
* the list [M, M+1, ... ,N] if M <= N;
* the empty list, [], otherwise.
If x and y are indeterminates in a ring, then
x .. y
gives the indeterminates between x and y in the order they appear
in the definition of the ring.
>EXAMPLE<
1..10;
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
-------------------------------
Use R ::= Q[x,y,z,a,b,c,d];
z..c;
[z, a, b, c]
-------------------------------
2.4 Evaluation and Assignment
2.4.1 Evaluation
An expression is by itself a valid command. The effect of this command is that the expression is evaluated in the current ring and its value is displayed. The evaluation of an expression in CoCoA is normally performed in a full recursive evaluation mode. Usually the result is the fully evaluated expression. The result of the evaluation is automatically stored in the variable It. >EXAMPLE< 2 + 2; 4 ------------------------------- It + 3; 7 ------------------------------- It; 7 ------------------------------- X := 5; It; 7 ------------------------------- The command 'X := 5' is an assignment, not an evaluation; so it does not change the value of the variable It. If an error occurs during the evaluation of an expression, then the evaluation is interrupted and the user is notified about the error.
2.4.2 Assignment
An assignment command has the form L := E where L is a variable and E is an expression. The assignment command binds the result of the evaluation of the expression E to L in the working memory (see the chapter entitled "Memory Management"). If E is dependent upon a ring, then L is labeled with that ring. The label is listed when L is evaluated in another ring. Then command 'RingEnv(L)' will return the label for L. Use R ::= Q[t,x,y,z]; I := Ideal(x,y); M := 5; N := 8; T := M+N; T; 13 ------------------------------- T := T+1; -- note that T occurs on the right, also T; 14 ------------------------------- L := [1,2,3]; L[2] := L[3]; L; [1, 3, 3] ------------------------------- P := Record[F = xz]; P.Degree := Deg(P.F); P; Record[Degree = 2, F = xz] ------------------------------- Use S ::= Q[a,b]; I; -- I is labeled by R since it depends on R R :: Ideal(x, y) ------------------------------- T; -- T is not labeled by R 14 ------------------------------- J := R:: Ideal(x^2-y); -- J contains an object dependent on R J; -- since the ring S is active, J is labeled by R R :: Ideal(x^2 - y) ------------------------------- Use R; J; Ideal(x^2 - y) ------------------------------- For information about interacting with rings outside of the current ring, see 'Accessing Other Rings' in the chapter entitled "Rings". To assign values to global variables, see 'Introduction to Memory' or 'Global Memory'.
2.5 User-Defined Functions
2.5.1 Introduction to User-Defined Functions
The most important construct in CoCoA programming is the user-defined function. These functions take parameters, perform CoCoA commands, and return values. Collections of functions can be stored in text files and read into CoCoA sessions using 'Source' (or '<<'). To prevent name conflicts of the type that are likely to arise if functions are to be made available for use by others, the functions can be collected in "packages," as described in a later chapter. To learn about user functions, look up 'Define' (online, enter '?define') for the complete syntax and for examples. The tutorial also contains several examples of functions.
2.5.2 Commands and Functions for User-Defined Functions
User-defined functions can contain just about any CoCoA command and
may refer to other user-defined functions. The following are some
commands that pertain particularly to functions:
* Call -- apply a function to given arguments
* Define -- define a function
* Function -- return a function
* Functions -- list the functions of a package
* NewId -- create a new identifier
* Return -- exit from a structured command
* Var -- function calls by reference, other complex referencing
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
2.6 Flow Control: Conditional Statements and Loops
2.6.1 Commands and Functions for Branching
The following are the CoCoA commands for constructing conditional
statements:
* Cond -- conditional expression
* If -- conditional statement
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
2.6.2 Commands and Functions for Loops
The following are the commands and functions for loops:
* Break -- break out of a loop
* For -- loop command
* Foreach -- loop command
* Repeat -- loop command
* Return -- exit from a structured command
* While -- loop command
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
2.7 Input/Output
2.7.1 Introduction to IO
Input and output is implemented in CoCoA through the use of "devices." At present, the official devices are: 1) standard IO (the CoCoA window), 2) text files, and 3) strings. What this means is that it is possible to read from or write to any of these places. The cases are discussed separately, below. Text files may be read verbatim or---with the 'Source' command---be executed as CoCoA commands.
2.7.2 Standard IO
Standard IO is what takes places normally when one interacts with
CoCoA via the CoCoA window. CoCoA accepts and interprets strings
typed in by the user and prints out expressions. If E is a CoCoA
object, then the command
E;
causes the value of E to be printed to the CoCoA window. One may also
use the functions 'Print' and 'PrintLn' for more control over the
format of the output.
The official devices that are being used here are 'DEV.STDIN' and
'DEV.OUT'. So for instance, the commands 'Get' and 'Print On' can be
used with the standard devices although they are really meant to be
used with the other devices. 'Print E On DEV.OUT' is synonymous with
'Print E'. Also, one may use 'Get(DEV.STDIN,10)', for example, to get
the next 10 characters typed in the CoCoA window. Thus, clever use of
'Get' will allow your user-defined functions to prompt the user for
input, but normal practice is to pass variables to a function as
arguments to that function.
2.7.3 File IO
To print CoCoA output to a file, one first opens the file
with 'OpenOFile' then prints to the file using 'Print On'.
To receive verbatim input from a file, one first opens the file with
'OpenIFile', then gets characters from the file with 'Get'. Actually,
'Get' gets a list of ascii codes for the characters in the file.
These can be converted to real characters using the function 'Ascii'.
>EXAMPLE<
D := OpenOFile('my-file'); -- open text file with name 'my-file',
-- creating it if necessary
Print 'hello world' On D; -- append 'hello world' to my-file
Close(D); -- close the file
D := OpenIFile('my-file'); -- open 'my-file'
Get(D,10); -- get the first ten characters, in ascii code
[104, 101, 108, 108, 111, 32, 119, 111, 114, 108]
-------------------------------
Ascii(It); -- convert the ascii code
hello worl
-------------------------------
Close(D);
To read and execute a sequence of CoCoA commands from a text file, one
uses the 'Source' command, or equivalently '<<'. For instance, if the
file 'MyFile.coc' contains a list of CoCoA commands, then
<<'MyFile.coc';
reads and executes the commands.
See also:
Ascii
Close
Get
OpenIFile, OpenOFile
OpenLog
Print On
Source
2.7.4 String IO
To print CoCoA output to a string, on may use 'OpenOString' to
"open" the string, then 'Print On' to write to it. To read from a
string, one may open the string for input with 'OpenIString' then get
characters from it with 'Get'.
>EXAMPLE<
S := 'hello world';
D := OpenIString('',S); -- open the string S for input to CoCoA
-- the first argument is just a name for the device
L:= Get(D,7); -- read 7 characters from the string
L; -- ascii code
[104, 101, 108, 108, 111, 32, 119]
-------------------------------
Ascii(L); -- convert ascii code to characters
hello w
-------------------------------
Close(D); -- close device D
D := OpenOString(''); -- open a string for output from CoCoA
L := [1,2,3]; -- a list
Print L On D; -- print to D
D;
Record[Name = "", Type = "OString", Protocol = "CoCoAL"]
-------------------------------
S := Cast(D,STRING); -- S is the string output printed on D
S; -- a string
[1, 2, 3]
Print ' more characters' On D; -- append to the existing output string
Cast(D,STRING);
[1, 2, 3] more characters
-------------------------------
There are usually more direct ways to collect results in strings. For
instance, if the output of a CoCoA command is not already of type
STRING, one may convert it to a string using 'Sprint'.
2.7.5 Commands and Functions for IO
The following are commands and functions for input/output:
* Block -- group several commands into a single command
* Close -- close a device
* Format -- convert object to formatted string
* Get -- read characters from a device
* IO.SprintTrunc -- convert to a string and truncate
* Latex -- LaTeX formatting
* More -- print a string, N lines at a time
* OpenIFile, OpenOFile -- open input or output file
* OpenIString, OpenOString -- open input or output string
* OpenLog, CloseLog -- open or close a log of a CoCoA session
* Print On -- print to an output device
* Print, PrintLn -- print the value of an expression
* Source, << -- read commands from a file
* Sprint -- convert to a string
* Tag -- returns the tag string of an object
* Tagged, Untagged, @ -- tag or untag an object for pretty printing
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
2.7.6 Tagged Printing
Some CoCoA objects are intrinsically complicated, so printing them verbatim might be confusing. For this reason a mechanism has been implemented which enables automatic pretty printing through the use of "tags." A user may "tag" any object with a string and then define how objects tagged with that string should be printed or described. Commands that do not have to do with printing ignore the tag.
2.7.7 Tagging an Object
If E is any CoCoA object and S a string, then the function
'Tagged(E,S)' returns the object E tagged with the string S. The type
of the returned object is 'TAGGED(S)' (but the type of E is
unchanged). The function, 'Tag', returns the tag string of an object,
and the function 'Untagged' (or '@') returns the object, stripped of
its tag.
>EXAMPLE<
L := ['Dave','March 14, 1959',372];
M := Tagged(L,'MiscData'); -- L has been tagged with the string 'MiscData'
Type(L); -- L is a list
LIST
-------------------------------
Type(M); -- M is a tagged object
TAGGED("MiscData")
-------------------------------
Tag(M); -- the tag string of M (it would be the empty string if M
-- where not a tagged object).
MiscData
-------------------------------
M; -- Until a special print function is defined, the printing of L
-- and M is identical.
["Dave", "March 14, 1959", 372]
-------------------------------
Untagged(M) = L; -- 'Untagged' removes the tag from M, recovering L.
TRUE
-------------------------------
The next section explains how to define functions for pretty printing
of tagged objects.
2.7.8 Printing a Tagged Object
Suppose the object E is tagged with the string S. When one tries to print E---say with 'Print E' or just 'E;'--- CoCoA looks for a user-defined function with name 'Print_S'. If no such function is available, CoCoA prints E as if it were not tagged, otherwise, it executes 'Print_S'. >EXAMPLE< L := ['Dave','March 14, 1959',372]; -- continuing with the previous example M := Tagged(L,'MiscData'); M; -- M is printed as normal in the absence of a function 'Print_MiscData' ["Dave", "March 14, 1959", 372] ------------------------------- Define Print_MiscData(X) -- Exactly one parameter is required. M := Untagged(X); Print(M[1]); EndDefine; Print M; -- Now, any object tagged with the string 'MiscData' will be -- printed using Print_MiscData Dave ------------------------------- M; -- Whenever printing of M is called for, 'Print_MiscData' is executed. Dave ------------------------------- The line 'M := Untagged(X)' is actually not necessary here, but in general one may get into an infinite loop trying to print X, a tagged object, from within the function that is being defined in order to print X, if that makes sense. Untagging X prevents this problem.
2.7.9 Describing a Tagged Object
If the object E is tagged with the string S, then when the user
enters the command 'Describe E', CoCoA first looks for a user-defined
function with name 'Describe_S' and executes it; if not found, the
output depends on the type of Untagged(E).
>EXAMPLE<
Use R ::= Q[x,y,z];
I := Ideal(x-y^2,x-z^3);
I := Tagged(I,'MyIdeals'); -- I is now tagged with 'MyIdeals'
Describe I; -- the default description of an ideal
Record[Type = IDEAL, Value = Record[Gens = [-y^2 + x, -z^3 + x]]]
-------------------------------
Define Describe_MyIdeals(X)
Y := Untagged(X);
PrintLn('The generators are:');
Foreach G In Y.Gens Do
PrintLn(' ',G);
EndForeach;
EndDefine;
Describe I; -- Any object tagged with 'MyIdeals' is now described
-- using 'Describe_MyIdeals'.
The generators are:
-y^2 + x
-z^3 + x
-------------------------------
2.7.10 Another Example Using Tags
Here is one more example using tags. Note that CoCoA commands that
do not have to do with printing ignore tags.
>EXAMPLE<
N := Tagged(4,'Dots');
N;
4
-------------------------------
Define Print_Dots(X)
For I := 1 To X Do
Print '.'
EndFor
EndDefine;
N;
....
-------------------------------
N+N; -- As long as printing is not involved, N is treated as if
-- it has no tag. In this case, the sum of two tagged objects
-- returns an integer, not another tagged object.
8
-------------------------------
M := Tagged(12,'Dots');
M;
............
-------------------------------
2.7.11 Commands and Functions for Tags
The following are commands and functions involving tags:
* Tag -- returns the tag string of an object
* Tagged, Untagged, @ -- tag or untag an object for pretty printing
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
2.8 Memory Management
- Introduction to Memory
- Working Memory
- Global Memory
- Ring-Bound Memory
- Commands and Functions for Memory
2.8.1 Introduction to Memory
CoCoA has three types of memory: "working", "global", and "ring-bound" memory. Unlike previous versions of CoCoA, starting with CoCoA 3.5, variables defined during a session are by default assigned to a *working memory*, accessible from all rings (but not from user-defined functions). There are no longer variables that are local to a particular ring. However, as in previous versions of CoCoA, one may define variables in the *global memory* by using the prefix 'MEMORY'. The word 'global' now refers to the fact that these variables are accessible not only to all rings but also to user-defined functions. A special class of global variables can be stored in what is called the *ring-bound memory*. These variables are formed with the prefix 'MEMORY.ENV.R' where 'R' is a ring identifier; they are 'bound' to the ring, which means that they are automatically destroyed when their corresponding rings cease to exist. Otherwise, variables in the ring-bound memory behave exactly as all other global variables. Most users will never need the ring-bound memory. These three types of memory are discussed separately, below.
2.8.2 Working Memory
The *working memory* consists of all variables except those defined
with the prefix 'MEMORY', e.g. 'MEMORY.X'. All variables in the
working memory are accessible from all rings, but they are not
accessible from within a user-define function (see examples in the
next section). The function 'Memory' displays the contents of the
working memory. More information is provided by 'Describe Memory()'.
Ring-dependent variables such as those containing polynomials, ideals,
or modules, are labeled by their corresponding rings. If the ring of
a ring-dependent variable in the working memory is destroyed, the
variable will continue to exist, but labeled by a ring automatically
generated by CoCoA. Once all variables dependent on this new ring
cease to exist, so does the ring.
>EXAMPLE<
Use R ::= Q[x,y,z];
Memory(); -- the working memory is empty
[ ]
-------------------------------
I:= Ideal(xy-z^3,x^2-yz);
X := 3;
M := Mat([[1,2],[3,4]]);
Memory();
["I", "It", "M", "X"]
-------------------------------
Describe Memory();
------------[Memory]-----------
I = Ideal(-z^3 + xy, x^2 - yz)
It = ["I", "It", "M", "X"]
M = Mat[
[1, 2],
[3, 4]
]
X = 3
-------------------------------
Use S ::= Z/(3)[t]; -- switch to a different ring
X := t^2+t+1; -- the identifier X is used again
Y := 7;
Describe Memory(); -- note that I is labeled by its ring
------------[Memory]-----------
I = R :: Ideal(-z^3 + xy, x^2 - yz)
It = ["I", "It", "M", "X"]
M = Mat[
[1, 2],
[3, 4]
]
X = t^2 + t + 1
Y = 7
-------------------------------
GBasis(I); -- The Groebner basis for the ideal in R can be calculated
-- even though the current ring is S.
[R :: x^2 - yz, R :: -z^3 + xy]
-------------------------------
M^2;
Mat[
[7, 10],
[15, 22]
]
-------------------------------
Use R ::= Q[s,t]; -- redefine the ring R
I; -- Note that I is labeled by a new ring, automatically produced by
-- CoCoA. This ring will automatically cease to exist when there
-- are no longer variables dependent upon it, as shown below.
R#17 :: Ideal(-z^3 + xy, x^2 - yz)
-------------------------------
RingEnvs();
["Q", "Qt", "R", "R#17", "S", "Z"]
-------------------------------
I:=3; -- I is overwritten with an integer, and since it is the only
-- variable dependent on R#17, the ring R#17 ceases to exist.
RingEnvs(); -- Since the only variable that was dependent upon the
-- temporary ring 'R#17' was overwritten, that ring is
-- destroyed.
["Q", "Qt", "R", "S", "Z"]
-------------------------------
See also:
Commands and Functions for Memory
2.8.3 Global Memory
Starting with CoCoA 3.5, a "global" variable is one that is
accessible from within a user-defined function. A global variable is
formed by using the prefix 'MEMORY'. The special prefixes 'DEV',
'ENV', 'ERR', and 'PKG' are shorthand for 'MEMORY.DEV', 'MEMORY.ENV',
etc. Any global variable prefixed by 'MEMORY.ENV.R' where 'R' is the
identifier of a ring, becomes part of the ring-bound memory discussed
in the next section. A list of the global variables which are not
ring-bound is provided by the function 'GlobalMemory'.
>EXAMPLE<
Use R ::= Q[x,y,z];
X := 5; -- a variable called 'X' in the working memory
MEMORY.X := 7; -- a global variable
X;
5
-------------------------------
MEMORY.X;
7
-------------------------------
Memory(); -- the working memory
["It", "X"]
-------------------------------
GlobalMemory(); -- the global memory
["DEV", "ENV", "ERR", "PKG", "X"]
-------------------------------
Define Test()
PrintLn(MEMORY.X);
MEMORY.X := 'a new value';
PrintLn(X);
EndDefine;
-- MEMORY.X is accessible from within a function
-- X is not accessible within a function (thus we get an error)
Test();
7
-------------------------------
ERROR: Undefined variable X
CONTEXT: PrintLn(X)
-------------------------------
MEMORY.X; -- the contents of the global memory can be changed from
-- within a function
a new value
-------------------------------
Fields(MEMORY.ENV); -- a list of all defined rings
["Q", "Qt", "R", "Z"]
-------------------------------
2.8.4 Ring-Bound Memory
A variable prefixed by 'MEMORY.ENV.R' where 'R' is the identifier of
a ring, becomes bound to the ring R. This means that when R ceases to
exist, so does the variable (unlike variables that are part of the
working memory, labeled by R: see "Working Memory", above). The
collection of these variables comprises the *ring-bound memory.*
The variables bound to a ring R can be listed with the command
'Memory(R)'. Note that since ring-bound variables are, in particular,
prefixed by 'MEMORY', they are also part of the global memory
discussed in the previous section.
Most users will never need ring-bound variables. Their main use is
within functions which need to define and use rings temporarily,
destroying them (along with their variables) before returning.
The prefix 'ENV' is shorthand for 'MEMORY.ENV'.
>EXAMPLE<
Use R ::= Q[x,y,z];
X := Ideal(x,y); -- a variable in the current memory
ENV.R.Y := 'bound to R'; -- a variable in the memory bound to R
Use S ::= Q[a,b];
Z := 6;
Memory(); -- the working memory
["X", "Z"]
-------------------------------
Memory(R); -- the memory bound to R
["Y"]
-------------------------------
Destroy R;
Memory();
["It", "X", "Z"]
-------------------------------
X; -- Since X is not in the ring-bound memory of R,
-- it is not destroyed. It was *dependent* on the ring R,
-- so the base ring of R has been given the new name R#5.
R#5 :: Ideal(x, y)
-------------------------------
ENV.R.Y; -- this variable was destroyed along with R
ERROR: Unknown record field R
CONTEXT: ENV.R.Y
-------------------------------
RingEnvs();
["Q", "Qt", "R#5", "S", "Z"]
-------------------------------
2.8.5 Commands and Functions for Memory
The following are commands and functions for memory:
* Clear -- clear the working memory or a ring-bound memory
* Defined -- check if an expression has been defined
* Delete -- delete variables from the working memory
* Destroy -- delete rings
* GlobalMemory -- contents of global memory
* Memory -- contents of local memory
* RingEnvs -- names of all defined rings
* Size -- the amount of memory used by an object
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
2.9 CoCoA Packages
- Introduction to Packages
- First Example of a Package
- Package Essentials
- Package Sourcing and Autoloading
- Global Aliases
- Local Aliases
- More Examples of Packages
- Package Initialization
- Sharing Your Package
- Commands and Functions for Packages
- Supported Packages
2.9.1 Introduction to Packages
User-defined functions may be saved in separate files and read into a CoCoA session using the 'Source' command. If one sources several such files or, especially, if a file is to be made available for general use, a possible problem arises from conflicting function names. If two functions with the same name are read into a CoCoA session, only the one last read survives. To avoid this, functions may be collected in 'packages'. A CoCoA package is essentially a list of functions (made using the 'Define' command), labeled with a long prefix. A function from a package is referred to by the package prefix plus the function name. The user may type the full prefix, but the usual method is to create a short alias for the prefix. Details are provided below, starting with a short example. See also: Define Source
2.9.2 First Example of a Package
The following is an example of a package. It could be typed into a
window as-is during a CoCoA session, but we will assume that it is
stored in a file in the CoCoA directory under the name "one.pkg."
>SAMPLE PACKAGE<
Package $contrib/toypackage
Define IsOne(N)
If N = 1 Then Return TRUE Else Return FALSE EndIf;
EndDefine;
Define Test(N)
If $.IsOne(N) Then
Print 'The number 1.'
Else
Print 'Not the number 1.'
EndIf;
EndDefine;
EndPackage; -- of toypackage
=====================================
Below is output from a CoCoA session in which this package was used:
>EXAMPLE<
<<'one.pkg' -- read in the package
Test(1); -- error here because the function 'Test' is not defined
-------------------------------
ERROR: Unknown operator Test
CONTEXT: Test(1)
-------------------------------
$contrib/toypackage.Test(1); -- this is the name of the function
-- we are looking for
The number 1.
-------------------------------
Alias Toy := $contrib/toypackage; -- use an alias to save typing
Toy.Test(3);
Not the number 1.
-------------------------------
Toy.IsOne(3);
FALSE
-------------------------------
Once the package is read, the user can choose a "substitute prefix"
using the 'Alias' command and in that way avoid conflicts between
functions in various packages and save on typing.
Note one other thing: the function 'IsOne' is used in the definition
of 'Test'. In that case, it is referred to as '$.IsOne'. Otherwise,
CoCoA would look for a global function, outside of the package, called
'IsOne'. Forgetting this kind of reference is a common source of
errors when constructing a package.
2.9.3 Package Essentials
A package begins with
Package $PackageName
and ends with
EndPackage;
PackageName is a string that will be used to identify the package.
The dollar sign is required. There are no restrictions on the string
PackageName, but keep in mind that it serves to distinguish functions
in the package from those in all other CoCoA packages. A name of the
form "contrib/subject" is typical.
In between the 'Package' declaration and its 'EndPackage' one may: (1)
declare Aliases (see below), (2) define functions, and (3) make
comments (please).
If a function F in the package appears in the definition of another
function within the package, it must be referred to as $.F (or
$PackageName.F, or using a local alias, see below).
Typically, the user will read in the package using the 'Source'
command (or '<<'). After that, to save on typing, the user will
choose a global alias with which to refer to the package using the
syntax:
Alias ShortName := $PackageName;
where ShortName is any convenient string, hopefully not conflicting
with other global aliases. (A list of the global aliases is returned
by the function 'Aliases'.)
A package function, F, is then called using the name 'ShortName.F'.
2.9.4 Package Sourcing and Autoloading
As mentioned above, packages are usually saved in files and then read
into a CoCoA session using the command 'Source' (or '<<').
I. Full path name, ordinary file sourcing.
package name: $mypackage
file name: this/is/my/stuff.pkg
Suppose the name of your package is '$mypackage' and is kept in the
file with full pathname 'this/is/my/stuff.pkg'. Then the package can
be loaded into the session as usual with either of the commands:
Source('this/is/my/stuff.pkg');
<<'this/is/my/stuff.pkg';
Functions can then be called from the package using the package name,
$mypackage, as a prefix (or using aliases).
II. The standard package path, $-shortcut.
package name: $mypackage
file name: lib/mypackages/stuff.pkg (relative to cocoa directory)
A package is in the 'standard package path' if its file is kept in the
'lib' directory inside the cocoa directory. Suppose your package has
name '$mypackage' and is kept in the file with pathname (relative to
the cocoa directory) 'lib/mypackages/stuff.pkg'. Then the package can
be read by passing this pathname to 'Source', as above, but there are
the following shortcuts:
Source('$mypackages/stuff');
<<'$mypackages/stuff';
<<$mypackages/stuff -- quotes are optional in this case
In other words, the prefix '$' is taking the place of 'lib/' and the
suffix '.pkg' is (and must be) left off. Functions can then be called
as in the previous case.
III. Autoloading.
package name: $mypackages/stuff
file name: lib/mypackages/stuff.pkg (relative to cocoa directory)
Now suppose that the package is in the standard package path, as
above, in the file with pathname (relative to the cocoa directory)
'lib/mypackages/stuff.pkg'. However, now assume that the name of the
package is '$mypackages/stuff', i.e., that it matches the name of its
file (without 'lib/' and '.pkg'). Then, if any function from the
package is called, say '$mypackages/stuff.MyFunction', the package
will automatically be loaded. Of course, one may also source the
package using either method I or II, from above.
* Initialize *
NOTE: As explained in the section entitled "Package Intitialization,"
below, no matter which method is used to source a package, any
function in the package named 'Initialize' will automatically be
executed when the package is loaded.
2.9.5 Global Aliases
A global alias for a package is formed by using the command 'Alias'
during a CoCoA session. (Local aliases are formed with the same
command, but are declared inside a package. They are for use only
within the package.) The syntax for 'Alias' is
Alias binding, ..., binding;
where a 'binding' has the form
identifier := $PackageName
The function 'Aliases' prints a list of the global aliases.
>EXAMPLE<
Aliases();
H = $cocoa/help
IO = $cocoa/io
GB = $cocoa/gb
HP = $cocoa/hp
HL = $cocoa/hilop
List = $cocoa/list
Mat = $cocoa/mat
Latex = $cocoa/latex
LaTeX = $cocoa/latex
Toric = $cocoa/toric
Coclib = $cocoa/coclib
TT = $abc
-------------------------------
Alias My := $my_package,
Old := $my_package/old_version;
Aliases();
HP = $cocoa/hp
BinRepr = $cocoa/binrepr
SpPoly = $cocoa/sppoly
HL = $cocoa/hilop
H = $cocoa/help
My = $my_package
Old = $my_package/old_version
-------------------------------
Note: global aliases cannot be used in function definitions. This is
to force independence of context. Inside a function, one must use the
complete package name. For example, '$cocoa/gb.Step(M)' is a
valid statement inside a function, but not 'GB.Step(M)'.
See also:
Alias
Aliases
2.9.6 Local Aliases
A local alias is an alias declared inside a package, for use only within the package. A local alias can have the same identifier as a global alias. Only local aliases are recognized within packages. There are two uses for local aliases. First, recall that if the definition of function in a package uses another function, F, also defined in the package, then F must be referred to using the package name as a prefix or, for short, '$.F'. In this way, '$', is an automatically a local alias for the package itself. One may choose another alias, say 'DD', and write 'DD.F', instead. A second use for a local alias is to refer to a separate package. In that way, one may refer to functions from that package inside the current package without typing out the full package name. Keep in mind that these aliases are used only to save typing. Examples appear below.
2.9.7 More Examples of Packages
Here is a simple package for printing lists.
>EXAMPLE<
Package $contrib/list
Define About()
Return "
Author: Antonio
Version: 1.0
Date: 18 July 1997
"
EndDefine;
Define PrintList(L)
Foreach X In L Do
PrintLn X
EndForeach
EndDefine;
EndPackage;
=================================
Here is another package that takes a pair of objects and turns the
pair into a list. Note the local alias used to reference the previous
package.
Package $contrib/pair
Alias L := $contrib/list; -- Local alias for another package.
-- This alias does not affect global
-- aliases.
Define Make(A,B)
Return [A,B];
EndDefine;
Define First(P)
Return P[1];
EndDefine;
Define Second(P)
Return P[2];
EndDefine;
Define PrintPairOfLists(P)
PrintLn 'First list:';
L.PrintList($.First(P)); -- The local alias, L, is used here,
PrintLn 'Second list:';
L.PrintList($.Second(P)) -- and here. '$' refers to a function
EndDefine; -- defined in the current package.
EndPackage;
=========================
USING THE PACKAGES. After reading in the packages using 'Source' or
'<<', one may proceed as follows to use them:
>EXAMPLE<
Alias P := $contrib/pair;
X := P.Make([x^2,x],[x,y,z]);
P.PrintPairOfLists(X);
First list:
x^2
x
Second list:
x
y
z
-------------------------------
Note: suppose a package with identifier $contrib/newlist prints lists
in another format. To switch to this format in $contrib/pair, one
need only change the alias for L from $contrib/list to
$contrib/newlist.
2.9.8 Package Initialization
Packages are often stored in files and read into a CoCoA session
using the 'Source' command. When a package is loaded, any function
whose name is 'Initialize' will automatically be executed. For
example, this feature may be used to print a message or initialized
global variables when the package is read into a session.
(Note: following this rule, if the first time you you access package
PKG is to make an explicit call to the function PKG.Initialize() then
the function will be called twice!)
>EXAMPLE<
Package $example
Define Initialize()
Print 'CALLED INITIALIZE';
MEMORY.MyVar := 17;
EndDefine;
EndPackage;
CALLED INITIALIZE
-------------------------------
MEMORY.MyVar;
17
-------------------------------
See also:
User Initialization
2.9.9 Sharing Your Package
If you create a package that others might find useful, please contact
the CoCoA team by email at cocoa@dima.unige.it.
Include comments in the package that:
* explain the use of the package
* list functions that are meant to be seen by the user
* give the syntax for these functions. (What are the arguments?
What values are returned?)
* describe what each function does
* provide examples of the use of each function.
2.9.10 Commands and Functions for Packages
The following are commands and functions for packages:
* Alias -- define aliases for package names
* Alias In -- temporarily override global aliases
* Aliases -- list of global aliases
* Functions -- list the functions of a package
* Packages -- list of loaded packages
* PkgName -- returns the name of a package
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
2.9.11 Supported Packages
Several packages are supported by the CoCoA team. These packages
contain functions that are not built into CoCoA because they are of a
more specialized or experimental nature.
The supported packages are:
algmorph.pkg -- K-algebra homomorphisms
ext.pkg -- ext modules, depth, and presentations
galois.pkg -- computing in a cyclic extension
intprog.pkg -- integer programming
invariants.pkg -- generators of an algebra of invariants
primary.pkg -- primary ideals
specvar.pkg -- special varieties
stat.pkg -- statistics, design of experiment
thmproving.pkg -- geometricla theorem proving
typevectors.pkg -- typevectors for ideals of points
All of these packages are included in /lib/contrib of the distribution
of CoCoA. The packages are likely to be updated more often than
CoCoA, itself, and new packages may appear; so it may be worth
checking at the CoCoA distribution sites, e.g.,
http://cocoa.dima.unige.it/.
HOW TO USE A SUPPORTED PACKAGE
-- save the package in lib/contrib/, if necessary;
-- to get the syntax, description, and examples of the main functions
and a suggested alias for the package, type
$contrib/"package_name".Man();
-- to find the version number, type
$contrib/"package_name".About();
NOTE: The packages will load automatically when one of their functions
is called (see "Package Sourcing and Autoloading") for more
information.
See below for more details about specific supported packages.
2.9.12 K-Algebra Homomorphisms
Supported CoCoA Package
TITLE : algmorph.pkg
DESCRIPTION : CoCoA package for computing K-algebra homomorphisms
and subalgebras
AUTHOR : A. Bigatti
LOADING INSTRUCTIONS
-- Enter
$contrib/algmorph.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/algmorph.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.13 Ext Modules
Supported CoCoA Package
TITLE : ext.pkg
DESCRIPTION : CoCoA package for computing Ext modules, depth, and presentation
AUTHOR : A. Damiano
LOADING INSTRUCTIONS
-- Enter
$contrib/ext.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/ext.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.14 Galois Package
Supported CoCoA Package
TITLE : galois.pkg
DESCRIPTION : CoCoA package for computing in a cyclic algebraic
extension
AUTHOR : A. Bigatti, D.La Macchia, F.Rossi
LOADING INSTRUCTIONS
-- Enter
$contrib/galois.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/galois.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.15 Integer Programming
Supported CoCoA Package
TITLE : intprog.pkg
DESCRIPTION : CoCoA package for applying toric ideals to integer
programming
AUTHOR : A. Bigatti
LOADING INSTRUCTIONS
-- Enter
$contrib/intprog.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/intprog.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.16 Algebra of Invariants
Supported CoCoA Package
TITLE : invariants.pkg
DESCRIPTION : CoCoA package for computing homogeneous generators of an
algebra of invariants, and for testing invariance of a polynomial
AUTHOR : A. Del Padrone
LOADING INSTRUCTIONS
-- Enter
$contrib/invariants.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/invariants.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.17 Primary Ideals
Supported CoCoA Package
TITLE : primary.pkg
DESCRIPTION : CoCoA package for applying toric ideals to integer
programming
AUTHORS : A. Bigatti, L. Robbiano
LOADING INSTRUCTIONS
-- Enter
$contrib/primary.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/primary.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.18 Special Varieties
Supported CoCoA Package
TITLE : specvar.pkg
DESCRIPTION : CoCoA package for computating the Hilbert-Poincare
series of special varieties (Segre, Veronese, Rees).
AUTHORS : A. Bigatti, L. Robbiano
LOADING INSTRUCTIONS
-- Enter
$contrib/specvar.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/specvar.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.19 Statistics
Supported CoCoA Package
TITLE : stat.pkg
DESCRIPTION : package for design of experiments in statistics
AUTHOR : M. Caboara
LOADING INSTRUCTIONS
-- Enter
$contrib/stat.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/stat.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.20 Typevectors
Supported CoCoA Package
TITLE : thmproving.pkg
DESCRIPTION : CoCoA package for geometrical theorem-proving in euclidean space
AUTHOR : L. Bazzotti, G. Dalzotto
LOADING INSTRUCTIONS
-- Enter
$contrib/thmproving.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/thmproving.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
2.9.21 Geometrical Theorem-Proving
Supported CoCoA Package
TITLE : typevectors.pkg
DESCRIPTION : CoCoA package for computing typevectors associated to
Hilbert functions of ideals of points
AUTHOR : E.Carlini, M.Stewart
LOADING INSTRUCTIONS
-- Enter
$contrib/typevectors.Man();
to get a complete description of the package including a suggested
alias.
-- Enter
$contrib/typevectors.About();
to find the version number. You may want to check the CoCoA
homepage for the latest version.
Doing Mathematics with CoCoA
3.1 Booleans
3.1.1 Introduction to Booleans
The two boolean constants are TRUE and FALSE. They are mainly used
with the commands 'If' and 'While', etc., inside CoCoA programs. The
relational operators
= <> < <= > >= IsIn
return boolean constants (see 'Relational Operators'). The boolean
operators are
Not And Or
and return boolean constants and are described below.
See also:
Relational Operators
Boolean Operators
3.1.2 Commands and Functions for Booleans
The following are commands and functions for booleans:
* EqSet -- checks if the set of elements in two lists are equal
* IsEven, IsOdd -- test whether an integer is even, odd, or zero
* IsHomog -- test whether given polynomials are homogeneous
* IsIn -- check if one object is contained in another
* IsNumber -- checks if the argument is a number
* IsPosTo, IsToPos -- checks the module term-ordering of a ring
* IsZero -- test whether an object is zero
* Not, And, Or -- boolean operators
* SubSet -- checks if the elements of one list are a subset of another
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.2 Numbers
3.2.1 Introduction to Numbers
There are three types of numbers recognized by CoCoA: integers (type INT), rationals (type RAT), and modular integers (type ZMOD). Numbers in CoCoA are handled with arbitrary precision. This means that the sizes of numbers are only limited by the amount of available memory. The basic numeric operations---addition (+), subtraction (-), multiplication (*), division (/), exponentiation (^), and negation (-)---behave as one would expect. Be careful, two adjacent minus signs, '--', start a comment in CoCoA. >EXAMPLE< N := 3; -N; -3 ------------------------------- --N; The last line in the above example does not return 3; it is interpreted as a comment.
3.2.2 Rationals
Rational numbers can be entered as fractions or as terminating decimals. CoCoA always converts a rational number into a fraction in lowest terms. >EXAMPLE< 3/5; 3/5 ------------------------------- 3.8; 19/5 ------------------------------- N := 4/8; N; 1/2 -------------------------------
3.2.3 Numerators and Denominators for Rational Numbers
If F is a variable holding a fraction, then F.Num and F.Den are the numerator and denominator, respectively. The functions 'Num' and 'Den', respectively, return the same. >EXAMPLE< F := 3/5; F.Num; 3 ------------------------------- F.Den; 5 ------------------------------- Den(F); 5 ------------------------------- 1.75; 7/4 ------------------------------- It.Num; 7 ------------------------------- See also: Den Num
3.2.4 Modular Integers
Let A and B be integers. The expression A%B has type ZMOD and represents the class of A modulo B. The integer B should be greater than 0 and less then 32767 = 2^15 - 1. When a modular integer is evaluated by CoCoA, it is reduced to a canonical form A%B with -B/2 < A <= B/2. Two modular integers of the form A%C and B%C are said to be 'compatible', and the usual arithmetical operations are applicable. >EXAMPLE< 3%7; 3 % 7 ------------------------------- 4%7; -3 % 7 ------------------------------- 2%5 + 4%5; 1 % 5 ------------------------------- Type(3%11); ZMOD ------------------------------- 3%11 = 14%11; TRUE ------------------------------- 3%11 = 3; FALSE ------------------------------- Use the functions 'Div' and 'Mod' for quotients and remainders. See also: Div Mod
3.2.5 Commands and Functions for Numbers
The following are commands and functions for numbers:
INTEGERS
* Abs -- absolute value of a number
* Bin -- binomial coefficient
* BinExp, EvalBinExp -- binomial expansion
* Div, Mod -- quotient and remainder for integers
* EvalHilbertFn -- evaluate the Hilbert function
* Fact -- factorial function
* Fraction -- returns the quotient of its arguments
* GCD, LCM -- greatest common divisor, least common multiple
* GenericPoints -- random projective points
* Inverse -- multiplicative inverse
* IsEven, IsOdd -- test whether an integer is even or odd
* IsNumber -- checks if the argument is a number
* IsPrime -- prime integer test
* Isqrt -- computes the (truncated) square root of an integer
* IsZero -- test whether an object is zero
* Len -- the length of an object
* Mod2Rat -- reconstructing rationals from modular integers
* Max, Min -- a maximum or minimum element of a sequence or list
* NextPrime -- find the next largest prime number
* Num, Den -- numerator, denominator
* Product, Sum -- the product or sum of the elements of a list
* Rand -- random integer
* Seed -- seed for 'Rand'
* Size -- the amount of memory used by an object
RATIONALS
* Abs -- absolute value of a number
* FloatStr, MantissaAndExponent -- convert rational number to a float string
* Fraction -- returns the quotient of its arguments
* Inverse -- multiplicative inverse
* IsNumber -- checks if the argument is a number
* IsZero -- test whether an object is zero
* Mod2Rat -- reconstructing rationals from modular integers
* Max, Min -- a maximum or minimum element of a sequence or list
* Num, Den -- numerator, denominator
* Product, Sum -- the product or sum of the elements of a list
MODULAR INTEGERS
* Fraction -- returns the quotient of its arguments
* Inverse -- multiplicative inverse
* IsNumber -- checks if the argument is a number
* IsZero -- test whether an object is zero
* Mod2Rat -- reconstructing rationals from modular integers
* Product, Sum -- the product or sum of the elements of a list
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.3 Strings
3.3.1 Introduction to Strings
A string constant consists of a sequence of characters between quotes
(') or between double quotes (").
>EXAMPLE<
'This is a string.';
This is a string.
-------------------------------
"So is this.";
So is this.
-------------------------------
One may not, however, start a string with a single quote and end with
a double quote, or vice versa.
See also:
Concatenation
Concat
Substrings
Quotes Within Strings
3.3.2 Concatenation
Strings may be concatenated using '+' or using the list function 'Concat'. >EXAMPLE< L := 'hello '; M := 'world'; L + M; hello world ------------------------------- Concat(L,M); hello world ------------------------------- See also: Concat
3.3.3 Substrings
If L is a string and N is an integer, then L[N] is the N-th character
of L.
>EXAMPLE<
L := 'hello world';
L[2];
e
-------------------------------
The operator 'IsIn' can be used to test if one string is a substring
of another.
>EXAMPLE<
L := 'hello world';
'hello' IsIn L; -- one may also write IsIn('hello',L)
TRUE
-------------------------------
See also:
IsIn
3.3.4 Quotes Within Strings
Strings are delimited using single quotes or double quotes (but not mixed). One may directly use quotes inside a string if they are not of the same type as the delimiters. To get quotes inside a string which *are* of the same type as the delimiters, the quotes must be doubled. >EXAMPLE< 'This string ''contains'' quotes of "various" styles.'; -- CoCoA Input This string 'contains' quotes of "various" styles. ------------------------------- "This string also 'contains' quotes of ""various"" styles."; -- CoCoA Input This string also 'contains' quotes of "various" styles. ------------------------------- Imagine the difficulties in writing this section of the online manual within a CoCoA package. ;>
3.3.5 Commands and Functions for Strings
The following are commands and functions for strings:
* Ascii -- convert between characters and ascii code
* Comp -- the N-th component of a list
* FloatStr, MantissaAndExponent -- convert rational number to a float string
* Format -- convert object to formatted string
* IO.SprintTrunc -- convert to a string and truncate
* IsIn -- check if one object is contained in another
* Latex -- LaTeX formatting
* Max, Min -- a maximum or minimum element of a sequence or list
* More -- print a string, N lines at a time
* NewId -- create a new identifier
* OpenIString, OpenOString -- open input or output string
* Spaces -- return a string of spaces
* Sprint -- convert to a string
* Var -- function calls by reference, other complex referencing
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.4 Lists
3.4.1 Introduction to Lists
A CoCoA list is a sequence of CoCoA objects between brackets. In
particular, a list may contain other lists. The empty list is []. If
L is a list and N is an integer, then L[N] is the N-th component of
L. If L contains sublists, then
L[N_1,N_2,...,N_s]
is shorthand for L[N_1][N_2]...[N_s] (see the example below). Lists
are often used to build structured objects of type MAT, VECTOR, IDEAL,
and MODULE.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
L := [34x+y^2,'a string',[],[TRUE,FALSE]]; -- a list
L[1]; -- the list's 1st component
y^2 + 34x
-------------------------------
L[2];
a string
-------------------------------
L[3];
[ ]
-------------------------------
L[4]; -- The 4th component is a list, itself;
[TRUE, FALSE]
-------------------------------
L[4][1]; -- its 1st component;
TRUE
-------------------------------
L[4,1]; -- the same.
TRUE
-------------------------------
[1,"a"]+[2,"b"]; -- Note: one may add lists if their components are
[3, "ab"] -- compatible (see 'Algebraic Operators').
-------------------------------
L := [x^2-y,ty^2-z^3];
I := Ideal(L);
I;
Ideal(x^2 - y, ty^2 - z^3)
-------------------------------
3.4.2 Commands and Functions for Lists
CoCoA provides a variety of commands for manipulating lists. Note in
particular the command 'In' which is useful for building lists.
The following are commands and functions for lists:
* .. -- range operator
* >< -- Cartesian product
* Append -- append an object to an existing list
* BlockMatrix -- create a block matrix
* BringIn -- bring in objects from another ring
* Comp -- the N-th component of a list
* Concat, ConcatLists -- concatenate lists or lists of lists, respectively
* Count -- count the objects in a list
* Diff -- returns the difference between two lists
* Distrib -- the distribution of objects in a list
* EqSet -- checks if the set of elements in two lists are equal
* First -- the first N elements of a list
* Flatten -- flatten a list
* GBM, HGBM -- intersection of ideals for zero-dimensional schemes
* GenericPoints -- random projective points
* Head -- the first element of a list
* HIntersection, HIntersectionList -- intersection of ideals
* IdealAndSeparatorsOfPoints -- ideal & separators for affine points
* IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
* IdealOfPoints -- ideal of a set of affine points
* IdealOfProjectivePoints -- ideal of a set of projective points
* In -- create a list satisfying given conditions
* Insert, Remove -- insert or remove an object into a list
* Interpolate -- interpolating polynomial
* Interreduce, Interreduced -- interreduce a list of polynomials or vectors
* Intersection, IntersectionList -- intersect lists, ideals, or modules
* IsIn -- check if one object is contained in another
* Last -- the last N elements of a list
* Len -- the length of an object
* List -- convert an expression into a list
* Mat -- convert an expression into a matrix
* Max, Min -- a maximum or minimum element of a sequence or list
* Monic -- divide polynomials by their leading coefficients
* NewList -- create a new list
* NonZero -- remove zeroes from a list
* Product, Sum -- the product or sum of the elements of a list
* Reverse, Reversed -- reverse a list
* ScalarProduct -- scalar product
* SeparatorsOfPoints -- separators for affine points
* SeparatorsOfProjectivePoints -- separators for projective points
* Set -- remove duplicates from a list
* Size -- the amount of memory used by an object
* Sort, Sorted -- sort a list
* SortBy, SortedBy -- sort a list
* Submat -- submatrix
* SubSet -- checks if the elements of one list are a subset of another
* Syz -- syzygy modules
* Tail -- remove the first element of a list
* Toric -- saturate toric ideals
* Toric.CheckInput -- check input to 'Toric'
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.5 Records
3.5.1 Introduction to Records
A record is a data type in CoCoA representing a list of bindings of the form 'name --> object'. >EXAMPLE< Use R ::= Q[x,y,z]; P := Record[ I = Ideal(x,y^2-z), F = x^2 + y, Misc = [1,3,4]]; P.I; Ideal(x, y^2 - z) ------------------------------- P.F; x^2 + y ------------------------------- P.Misc; [1, 3, 4] ------------------------------- P.Misc[2]; 3 ------------------------------- P.Date := '1/1/98'; P; Record[Date = "1/1/98", F = x^2 + y, I = Ideal(x, y^2 - z), Misc = [1, 3, 4]] ------------------------------- P["I"]; -- equivalent to P.I Ideal(x, y^2 - z) ------------------------------- P["Misc",3]; -- equivalent to P.Misc[3] 4 ------------------------------- Each entry in a record is called a 'field'. Note that records are "open" in the sense that their fields can be extended, as shown in the previous example. At present, there is no function for deleting fields from a record: one must rewrite the record, selecting the fields to retain. >EXAMPLE< P := Record[A = 2, B = 3, C = 5, D = 7]; Q := Record[]; Foreach F In Fields(P) Do If F <> "C" Then Q.Var(F) := P.Var(F) EndIf; -- 'Q.F' would not work here EndForeach; P := Q; Delete Q; -- get rid of the variable Q P; Record[A = 2, B = 3, D = 7] -------------------------------
3.5.2 Commands and Functions for Records
The following are commands and functions for records:
* Comp -- the N-th component of a list
* DivAlg -- division algorithm
* Fields -- list the fields of a record
* IdealAndSeparatorsOfPoints -- ideal & separators for affine points
* IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
* Record -- create a record
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.6 Vectors
3.6.1 Introduction to Vectors
An object of type VECTOR in CoCoA represents a vector of
polynomials. If V is a vector and F is a polynomial, then the
following are also vectors:
+V, -V, F*V, V*F.
If V and W are vectors with the same number of components, then one
may add and subtract V and W componentwise:
V+W, V-W.
>EXAMPLE<
Use R ::= Q[x,y];
V := Vector(x+1,y,xy^2);
V;
Vector(x + 1, y, xy^2)
-------------------------------
-V;
Vector(-x - 1, -y, -xy^2)
-------------------------------
x*V;
Vector(x^2 + x, xy, x^2y^2)
-------------------------------
W := Vector(x,y,x^2y^2-y);
x*V-W;
Vector(x^2, xy - y, y)
-------------------------------
3.6.2 Commands and Functions for Vectors
The following are commands and functions for vectors:
* BringIn -- bring in objects from another ring
* CoeffOfTerm -- coefficient of a term of a polynomial or vector
* Colon, :, HColon -- ideal or module quotient
* ColumnVectors -- the list of column vectors of a matrix
* Comp -- the N-th component of a list
* Comps -- list of components of a vector
* Deg -- the degree of a polynomial or vector
* DivAlg -- division algorithm
* E_ -- canonical vector
* FirstNonZero, FirstNonZeroPos -- the first non-zero entry in a vector
* GenRepr -- representation in terms of generators
* IsIn -- check if one object is contained in another
* IsTerm -- checks if the argument is a term
* IsZero -- test whether an object is zero
* LC -- the leading coefficient of a polynomial or vector
* Len -- the length of an object
* List -- convert an expression into a list
* LM -- the leading monomial of a polynomial or vector
* LPos -- the position of the leading power-product in a vector
* LPP -- the leading power-product of a polynomial or vector
* LT -- the leading term of an object
* Mat -- convert an expression into a matrix
* Monomials -- the list of monomials of a polynomial or vector
* NewVector -- create a new vector
* NF -- normal form
* NFsAreZero -- test if normal forms are zero
* NonZero -- remove zeroes from a list
* NR -- normal remainder
* NumComps -- the number of components of a vector
* Product, Sum -- the product or sum of the elements of a list
* ScalarProduct -- scalar product
* Size -- the amount of memory used by an object
* Support -- the list of terms of a polynomial or vector
* Vector -- create a vector
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.7 Matrices
3.7.1 Introduction to Matrices
An m x n matrix is represented in CoCoA by the list of its rows
Mat(R_1,...,R_m)
where each R_i is of type LIST and has length n. A matrix has type
MAT. The (A,B)-th entry of a matrix M is given by M[A][B] or M[A,B].
>EXAMPLE<
Use R ::= Q[x,y,z];
M := Mat([[x,y,xy^2],[y,z^2,2+x]]);
M;
Mat[
[x, y, xy^2],
[y, z^2, x + 2]
]
-------------------------------
M[1][3];
xy^2
-------------------------------
M[1,3];
xy^2
-------------------------------
The following operations are defined as one would expect for matrices
M^A, +M, -N, M+N, M-N, M*N, F*M, M*F
where M,N are matrices, A is a non-negative integer, and F is a
polynomial or rational function, with the obvious restrictions on the
dimensions of the matrices involved.
>EXAMPLE<
Use R ::= Q[x,y];
N := Mat([[1,2],[3,4]]);
N^2;
Mat[
[7, 10],
[15, 22]
]
-------------------------------
x/y * N;
Mat[
[x/y, 2x/y],
[3x/y, 4x/y]
]
-------------------------------
N + Mat([[x,x],[y,y]]);
Mat[
[x + 1, x + 2],
[y + 3, y + 4]
]
-------------------------------
3.7.2 Commands and Functions for Matrices
The following are commands and functions for matrices:
* Adjoint -- adjoint matrix
* BlockMatrix -- create a block matrix
* BringIn -- bring in objects from another ring
* ColumnVectors -- the list of column vectors of a matrix
* DegLexMat, DegRevLexMat, LexMat, XelMat -- matrices for std. term-orderings
* Det -- the determinant of a matrix
* Identity -- the identity matrix
* Inverse -- multiplicative inverse
* IsZero -- test whether an object is zero
* Jacobian -- the Jacobian of a list of polynomials
* Len -- the length of an object
* LinKer -- find the kernel of a matrix
* LinSol -- find a solution to a linear system
* List -- convert an expression into a list
* Mat -- convert an expression into a matrix
* Minors -- list of minor determinants of a matrix
* NewMat -- create a new matrix
* Pfaffian -- the Pfaffian of a skew-symmetric matrix
* Product, Sum -- the product or sum of the elements of a list
* Size -- the amount of memory used by an object
* Submat -- submatrix
* TensorMat -- returns the tensor product of two matrices
* Toric -- saturate toric ideals
* Toric.CheckInput -- check input to 'Toric'
* Transposed -- the transposition of a matrix
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.8 Rings
- Introduction to Rings
- New Rings
- Accessing Other Rings
- Ring Mappings: the Image Function
- Quotient Rings
- Commands and Functions for Rings
3.8.1 Introduction to Rings
Polynomial rings play a central role in CoCoA. Indeed, every object in CoCoA is defined over a base ring which is a polynomial ring. The user can define many rings, but at any time a "current ring" is active within the system. Most commands use the current ring as the base ring. Once a ring has been defined, the system can handle the following mathematical objects defined over that ring: * numbers (integers, rationals, modular integers); * polynomials; * vectors of polynomials; * rational functions; * ideals; * modules (submodules of a free module); * lists of objects; * matrices of objects. Variables containing ring-dependent objects such as polynomials, ideals, and modules are "labeled" by their ring. Variables containing objects such as integers which are not dependent on a particular ring are not labeled. IMPORTANT NOTE: Starting with CoCoA 3.5, variables are no longer local to specific rings, i.e., all variables are accessible from all rings. The next sections explains how to create a ring.
3.8.2 New Rings
CoCoA starts with the default ring R = Q[t,x,y,z]. The command for
building a new ring is either:
I ::= C
or
I ::= C[X:INDETS], M_1, ... , M_n
where I is the identifier of a CoCoAL variable, C: RING is an
expression defining a coefficient ring (Z, Q or Z/(N)), X is an
expression that defines the indeterminates, and the M_i are
"modifiers." Each of these components is discussed in separate
sections, below.
The modifiers are used to modify the default settings of the base
ring. The modifiers are of three classes: term-orderings, weights,
term orderings for modules. These classes are discussed in separate
sections below. It is possible to have no modifiers. The default
values are: DegRevLex for the term-ordering, 1 for the weight of each
indeterminate, and ToPos for the module term-ordering.
After the ring is defined using the above syntax, it can be made to be
the current ring with the command 'Use' or it can be accessed
temporarily from within the current ring with the command
'Using'. See these two commands for more information. There are also
several examples of ring-building in the tutorial.
>EXAMPLE<
Use R ::= Q[xyz]; -- define and use the ring R
S ::= Z/(103)[xy], Lex; -- define the ring S with Lex term-ordering
CurrentRing(); -- the current ring is still R
Q[x,y,z]
-------------------------------
Use S; -- now the ring S is the current ring
Z/(103)[x,y]
-------------------------------
Using R Do X:=z^2-3 End; -- define a variable in R (not the current
-- ring)
Y := R::x^2+y^2+z^2; -- another way of defining a variable in R while
-- S is the current ring
T ::= Q[x[1..4]y[1..5,1..5]], Elim(y[1..5,1]), ToPos; -- a more
-- complicated ring
3.8.3 Coefficient Rings
As mentioned above, the coefficient ring for a CoCoA polynomial ring
may be:
1. Z: (arbitrarily large) integer numbers;
2. Q: (arbitrarily large) rational numbers;
3. Z/(N): (see 'CocoaLimits') integers modulo N,
The first two types of coefficients are based on the GNU-gmp library.
When integers modulo N are used, the system checks whether N is a
prime number and, if it is not, a warning message is given. However
non-prime integers are accepted. Hence it is possible to do some work
with polynomials whose coefficients are not in a field, but it is up
to the user to ensure that no illegal operation will be attempted. To
find the upper limit for the characteristic in the third case, see the
field 'MaxChar' returned by the function 'CocoaLimits'; N <= 32767 is
typical. (Note: 32003 is prime)
IMPORTANT NOTE: Presently the implementation of the Buchberger
algorithm for computing Groebner bases operates correctly only if
polynomials have coefficients in a field. So the high level operations
on ideals and modules (and the polynomial GCD) involving Groebner
bases computations work only if polynomials have coefficients in a
field. Otherwise it is very likely that the user will run into
trouble.
'CoeffRing': When creating a new ring, the word 'CoeffRing' may be
used to refer to the current coefficient ring. Examples below
illustrate its use.
>EXAMPLE<
Use R ::= Q[x,y]; -- coefficient ring is Q
S ::= Z/(5)[t]; -- coefficient ring is the field with 5 elements
T ::= Z[u,v]; -- A warning is issued if the coefficient ring
-- is not a field.
-- WARNING: Coeffs are not in a field
-- GBasis-related computations could fail to terminate or be wrong
-------------------------------
Use R ::= Z/(2)[x,y,z]; CurrentRing(); -- these examples show the usage
-- of 'CoeffRing'
Z/(2)[x,y,z]
-------------------------------
Use S ::= CoeffRing[a,b]; CurrentRing();
Z/(2)[a,b]
-------------------------------
3.8.4 Indeterminates
An "indeterminate" is represented by a name consisting of either a single lower case letter or a lower case letter followed by one or more indices. For example, x, x[1], x[1,2,3] are legal (and different) indeterminates, as is x[2I,2I+1] if I is an integer variable. When creating a ring, the indeterminates are listed, optionally separated by commas. Indeterminates with indices are formed with the syntax: x[R_1,...,R_n], where x is any lower case letter and each R_i has the form a_i..b_i for integers a_i <= b_i. >EXAMPLE< Use R ::= Q[x[1..2,4..8],y[1..3],u,v]; Indets(); [x[1,4], x[1,5], x[1,6], x[1,7], x[1,8], x[2,4], x[2,5], x[2,6], x[2,7], x[2,8], y[1], y[2], y[3], u, v] -------------------------------
3.8.5 Weights Modifier
In forming a ring, one of the possible modifiers that may be added has one of the forms: (i) 'Weights(W_1,...,W_n)' where W_i is a positive integer specifying the weight of the i-th indeterminate (the number of weights listed must be equal to the number of indeterminates) or (ii) Weights(M) where M is a matrix with as many columns as there are indeterminates. In the latter case, the i-th indeterminate has the multi-degree given by the i-th column of M. The first row of the matrix M must have all positive entries. The system defaults to the value of 1 for all indeterminates (in the absence of the Weights modifier). >EXAMPLE< Use S ::= Q[a,b,c], Weights(1,2,3); Deg(b); 2 ------------------------------- L := [1,2,3]; Use S ::= Q[a,b,c], Weights(L); Deg(b); 2 ------------------------------- W := Mat([[1,2,3],[4,5,6]]); Use S ::= Q[a,b,c], Weights(W); Deg(b); 2 ------------------------------- MDeg(b); -- the multi-degree of b [2, 5] ------------------------------- Deg(b^3+a^2c); 6 ------------------------------- MDeg(b^3+a^2c); [6, 15] ------------------------------- WeightsMatrix(); Mat[ [1, 2, 3], [4, 5, 6] ] ------------------------------- WeightsList(); -- returns the first row of the Weights Matrix [1, 2, 3] ------------------------------- See also: Deg MDeg WeightsList WeightsMatrix
3.8.6 Orderings
Polynomials are always sorted with respect to the ordering of their base ring. All the operations involving polynomials utilize and preserve this ordering. The user can define custom orderings or choose a predefined term-ordering. Some commands temporarily change the term-ordering in order to make a computation. The various possibilities are discussed more fully, below. The command 'Ord' can be used both as a modifier to set the ordering of the ring and as a way to view the matrix defining the current term-ordering. >EXAMPLE< Use R ::= Q[x,y,z], Lex; -- lexicographic term-ordering (predefined) Use S ::= Q[u,v], Ord([[1,1],[2,-3]]); -- custom term-ordering Ord(R); -- the matrix defining the term-ordering for R Mat[ [1, 0, 0], [0, 1, 0], [0, 0, 1] ] ------------------------------- Ord(S); -- the matrix defining the term-ordering for S Mat[ [1, 1], [2, -3] ] ------------------------------- See also: Ord
3.8.7 Predefined Term-Orderings
The predefined term-orderings are:
* degree reverse lexicographic: DegRevLex (the default ordering)
* degree lexicographic: DegLex
* pure lexicographic: Lex
* pure xel: Xel
* elimination term-ordering: Elim(X:INDETS)
The first two term-orderings use the weights of the indeterminates for
computing the degree of a monomial. If the indeterminates are given in
the order x_1, ..., x_n, then x_1 > ... > x_n with respect to Lex, but
x_1 < ... < x_n with respect to Xel.
In the last ordering, X specifies the variables that are to be
eliminated. It may be a single indeterminate or a range of
indeterminates. However, X may not be an arbitrary list of
indeterminates; for that, see the command 'Elim' (as opposed to the
modifier 'Elim' being discussed here). A range of indeterminates can
be specified using the syntax < first-indet >..< last-indet >.
Another shortcut: if there are indexed variables of the form x[i,j],
then 'Elim(x)' specifies a term-ordering for eliminating all of the
x[i,j].
>EXAMPLE<
Use R ::= Q[x,y,z], Lex;
x+y+z;
x + y + z
-------------------------------
Use R ::= Q[x,y,z], Xel;
x+y+z;
z + y + x
-------------------------------
Use R ::= Q[t,x,y,z], Elim(t);
I := Ideal(t-x,t-y^2,t^2-xz^3);
GBasis(I);
[t - x, -y^2 + x, xz^3 - x^2]
-------------------------------
Use R ::= Q[x[1..5],y,z], Elim(x); -- term-ordering for eliminating all
-- of the x[i,j]'s
Ord();
Mat[
[1, 1, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0, -1],
[0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, -1, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0, 0]
]
-------------------------------
See also:
Elim
3.8.8 Temporary Term-Orderings
TEMPORARY TERM-ORDERINGS. For computations which temporarily require a different term-ordering (for example, to eliminate variables or to homogenize ideals), the system automatically changes the term-ordering to a more suitable one, performs the computation, and then restores the initial term-ordering and gives its output with respect to this one. In this way the user never has to deal with temporary changes. >EXAMPLE< Use R ::= Q[t,x,y,z]; I := Ideal(t-x,y-z,t-z); Elim(y,I); Ideal(t - z, -x + z) ------------------------------- See also: Elim Homogenized
3.8.9 Custom Term-Orderings
CUSTOM ORDERINGS. For special purposes, the user can enter a custom
ordering. We recall that each ordering > on the set of the terms of a
polynomial ring in n variables corresponds to a (not uniquely
determined) array (u_1,...,u_s) of vectors of the real vector space
R^n. More precisely if a = (a_1, ... ,a_n) and b = (b_1, ... ,b_n)
are the n-tuples of the exponents of two terms t and t', then
t > t' <=> (a . u_1, ..., a . u_s) >_{lex} (b . u_1, ..., b . u_s)
where >_{lex} is the ordering on R^s given by: (c_1,...,c_s)>_{lex}
(d_1,...,d_s) if and only if the first leftmost non-zero coordinate of
(c_1-d_1,...,c_s-d_s) is positive.
CoCoA accepts any ordering that can be defined by means of an n x n
matrix of integers (this is not a real restriction if one is
interested, for instance, in finding all possible Groebner bases of a
given ideal).
Moreover > is a term-ordering if and only if the matrix whose rows are
the vectors (u_1,...,u_s) has maximal rank and is such that the first
non-zero element in each column is positive.
To compute a Groebner basis a term-ordering is needed.
>EXAMPLE<
-- The following CoCoA command defines S to be the polynomial ring and
-- orders the terms of S using the term-ordering corresponding to the
-- vectors (1,1,0,0),(0,-1,0,0),(0,0,1,1),(0,0,0,-1):
Use S ::= Q[x,y,z,t], Ord( Mat([[1,1,0,0],
[0,-1,0,0],
[0,0,1,1],
[0,0,0,-1]]));
See also:
Ord
3.8.10 Module Orderings
First we recall the definition of a module term-ordering. We assume
that all our free modules have finite rank and are of the type M = R^r
where R is a polynomial ring with n indeterminates. Let
{e_i|i=1,...,r} be the canonical basis of M. A "term" of M is an
element of the form Te_i where T belongs to the set T(R) of the terms
of R. Hence the set T(M), of the terms of M, is in one-to-one
correspondence with the Cartesian product, T(R)x{1,...,r}.
A "module term-ordering" is defined as a total ordering > on T(M)
such that for all T, T_1, T_2 in T(R), with T not equal to 1, and for
all i, j in {1,...,r},
* T T_1 e_i > T_1 e_i
* T_1 e_i > T_2 e_j => T T_1 e_i > T T_2 e_j
Each term-ordering on the current ring induces several term-orderings
on a free module. CoCoA allows the user to choose between the
following:
* the ordering called "ToPos" (which is the default one) defined by:
T_1 e_i > T_2 e_j <=> T_1 > T_2 in R
or, if T_1 = T_2 , i < j
* the ordering called "PosTo" defined by:
T_1 e_i > T_2 e_j <=> i < j
or, if i = j, T_1 > T_2 in R .
>EXAMPLE<
-- The leading term of the vector (x,y^2) with respect two different
-- module term-orderings:
Use R ::= Q[x,y],ToPos;
LT(Vector(x,y^2));
Vector(0, y^2)
-------------------------------
Use R ::= Q[x,y],PosTo;
LT(Vector(x,y^2));
Vector(x, 0)
-------------------------------
3.8.11 Accessing Other Rings
There are a variety of ways of interacting with a ring outside of the
current ring. First of all, unlike CoCoA 3.4, starting with CoCoA
3.5, variables are usually assigned to a "working memory" accessible
from all rings. (The only exceptions are variables prefixed by
'MEMORY'. See the chapter entitled "Memory Management" for further
information.) If a variable contains an object which does not depend
on a user-defined ring---for example an integer---that object can be
immediately accessed and used within any ring. If a variable contains
a ring-dependent object such as a polynomial, an ideal, or a module,
the variable becomes labeled by the ring in which it was defined.
Built-in CoCoA functions should be smart enough to take into account
the rings on which their arguments depend (if you find an exception,
please send a message to cocoa@dima.unige.it).
To access rings outside of the current ring, one may of course use the
command 'Use' to change the current ring. Some other ways of
interacting with outside rings:
1. The '::' construction. This construction can be used to define
variables or perform operations in rings ouside of the current ring.
>EXAMPLE<
Use R ::= Q[x,y,z];
I := Ideal(x,y,z)^3;
I;
Ideal(x^3, x^2y, x^2z, xy^2, xyz, xz^2, y^3, y^2z, yz^2, z^3)
-------------------------------
Use S ::= Z/(5)[a,b];
I; -- I is labeled by its ring, R
R :: Ideal(x^3, x^2y, x^2z, xy^2, xyz, xz^2, y^3, y^2z, yz^2, z^3)
-------------------------------
RingEnv(I); -- the name of the ring on which I is dependent
R
-------------------------------
R:: Poincare(R/I); -- To be sure, one may prefix any operation
-- on I by 'R::' although this should not
-- be necessary
(1 + 3a + 6a^2)
-------------------------------
R:: (x+y)^2; -- S is still the active ring, but we can perform
-- operations in R
R :: x^2 + 2xy + y^2
-------------------------------
J := R :: Ideal(x^2-y); -- while S is active, one may define an
-- object dependent on R. This variable
-- becomes part of the working memory.
J;
R :: Ideal(x^2 - y)
-------------------------------
Use R;
J; -- the label is not used if R is active
Ideal(x^2 - y)
-------------------------------
2. Using. From within the current ring one may temporarily perform
commands in an another ring using the command 'Using'. A brief
example appears below. For more information, see the online help
entry for 'Using'.
>EXAMPLE<
Use R ::= Q[x,y];
S ::= Z/(5)[a,b]; -- the current ring is still R
Using S Do
X := (a+b)^5; -- assign a value to a variable in another ring
EndUsing;
X;
S :: a^5 + b^5
-------------------------------
Use S;
X;
a^5 + b^5
-------------------------------
3. Image. To map objects from one ring to another, one may use the
command 'Image.' An introduction to this command appears in the
following section and more details can be found in the online help
entry, 'Image'.
4. QZP, ZPQ. The commands 'QZP' and 'ZPQ' can sometimes be used to
quickly map a polynomial or ideal from an outside ring into the
current ring. See the online help entry, 'QZP, ZPQ', for details.
See also:
Image
QZP, ZPQ
Use
Using
3.8.12 Ring Mappings: the Image Function
The function 'Image' implements a ring homomorphism. Suppose S is
the current ring and R is another ring. If X is an object in R, the
function 'Image' may be used to substitute polynomials in S for the
indeterminates in X. An example is given below and complete details
are given in the online help entry for 'Image'.
To make substitutions within a single ring, one would usually use
'Eval' or 'Subst' rather than 'Image'. To map a polynomial or ideal
from an outside ring into the current ring, the functions 'QZP' and
'ZPQ' are sometimes useful. To map a polynomial or rational funtion
(or a list, matrix, or vector of these) from R to S without changing
indeterminates, use the function 'BringIn'. ('BringIn' is only
applicable if the indeterminates of the object to be mapped are a
subset of those in S.)
>EXAMPLE<
Use R ::= Q[a,b,c];
X := a+b-3c;
Use S ::= Q[x,y];
F := RMap(x^2,2,y^2); -- syntax for defining a map: the n-th
-- indeterminate in the domain will be mapped to
-- the n-th element listed in RMap.
X; -- X lives in the ring R
R :: a + b - 3c
-------------------------------
Image(X,F); -- the image of E under the map F
x^2 - 3y^2 + 2
-------------------------------
Image(R:: (a+b)^2,F);
x^4 + 4x^2 + 4
-------------------------------
See also:
BringIn
Image
Eval
QZP, ZPQ
Subst
Using
3.8.13 Quotient Rings
If I is an ideal in a CoCoA ring R, then R/I represents the
corresponding quotient ring. It has type TAGGED("Quotient").
>EXAMPLE<
Use R ::= Q[x,y];
I := Ideal(y-x^2);
Q := R/I;
Hilbert(Q); -- the Hilbert function for Q
H(0) = 1
H(x) = 2 for x >= 1
-------------------------------
3.8.14 Commands and Functions for Rings
The following are commands and functions controlling rings:
* Characteristic -- the characteristic of a ring
* Clear -- clear the working memory or a ring-bound memory
* CurrentRing -- the current ring
* Deg -- the degree of a polynomial or vector
* Delete -- delete variables from the working memory
* Destroy -- delete rings
* Dim -- the dimension of a ring or quotient object
* Hilbert -- the Hilbert function
* HilbertPoly -- the Hilbert polynomial
* HVector -- the h-vector of a ring or quotient object
* Image -- ring homomorphism
* Indet -- individual indeterminates
* IndetInd -- the index of an indeterminate
* IndetIndex -- index of an indeterminate
* Indets -- list of current indeterminates
* IndetName -- the name of an indeterminate
* IsPosTo, IsToPos -- checks the module term-ordering of a ring
* MDeg -- multi-degree of an polynomial
* Multiplicity -- the multiplicity (degree) of a ring or quotient object
* NumIndets -- number of indeterminates
* Ord -- matrix defining a term-ordering
* Poincare, HilbertSeries -- the Poincare series
* QZP, ZPQ -- change field for polynomials and ideals
* Ring -- returns the ring with a given name
* RingEnv -- name of the current ring
* RingEnvs -- names of all defined rings
* TypeOfCoeffs -- type of the coefficients of the current ring
* Use -- command for making a ring active
* Using -- perform commands in non-active ring
* WeightsList -- first row of the weights matrix
* WeightsMatrix -- matrix of generalized weights for indeterminates
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.9 Polynomials
3.9.1 Introduction to Polynomials
An object of type POLY in CoCoA represents a polynomial. To fix terminology: a polynomial is a sum of terms; each term is the product of a coefficient and power-product, a power-product being a product of powers of indeterminates. (In English it is standard to use 'monomial' to mean a power-product, however, in other languages, such as Italian, monomial connotes a power product multiplied by a scalar. In the interest of world peace, we will use the term power-product in those cases where confusion may arise.) >EXAMPLE< The following are CoCoA polynomials: Use R ::= Q[x,y,z]; F := 3xyz + xy^2; F; xy^2 + 3xyz ------------------------------- Use R ::= Q[x[1..5]]; Sum([x[N]^2 | N In 1..5]); x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2 ------------------------------- CoCoA always keeps polynomials ordered with respect to the term-orderings of their corresponding rings.. The following algebraic operations on polynomials are supported: F^N, +F, -F, F*G, F/G if G divides F, F+G, F-G, where F, G are polynomials and N is an integer. The result may be a rational function. >EXAMPLE< Use R ::= Q[x,y,z]; F := x^2+xy; G := x; F/G; x + y ------------------------------- F/(x+z); (x^2 + xy)/(x + z) ------------------------------- F^2; x^4 + 2x^3y + x^2y^2 ------------------------------- F^(-1); 1/(x^2 + xy) -------------------------------
3.9.2 Evaluation of Polynomials
Polynomials may be evaluated using the function 'Subst'. More
generally, 'Subst' allows one to substitute polynomials from the
current ring for the indeterminates of a given polynomial. If
substitutions are to be made for each indeterminate, in order, it is
easier to use 'Eval'. For more general substitutions, see 'Image'.
>EXAMPLE<
Use R ::= Q[x,y,z];
F := x+y+z;
Eval(F,[2,1]); -- let x=2 and y=1 in F
z + 3
-------------------------------
Subst(F,[[x,2],[y,1]]); -- the same thing using 'Subst'
z + 3
-------------------------------
Subst(F,y,1); -- the syntax is easier when substituting for a single
-- indeterminate
x + z + 1
-------------------------------
Subst(F,[[y,x-y],[z,2]]); -- substitute x-y for y and 2 for z
2x - y + 2
-------------------------------
See also:
Eval
Image
Subst
3.9.3 Commands and Functions for Polynomials
The following are commands and functions for polynomials:
* Bin -- binomial coefficient
* BringIn -- bring in objects from another ring
* Coefficients -- list of coefficients of a polynomial or vector
* CoeffOfTerm -- coefficient of a term of a polynomial or vector
* Colon, :, HColon -- ideal or module quotient
* Deg -- the degree of a polynomial or vector
* DensePoly -- the sum of all power-products of a given degree
* Der -- the derivative of a rational function
* Discriminant -- the discriminant of a polynomial
* DivAlg -- division algorithm
* Eval -- substitute numbers or polynomials for indeterminates
* Factor -- factor a polynomial
* Fraction -- returns the quotient of its arguments
* GCD, LCM -- greatest common divisor, least common multiple
* GenRepr -- representation in terms of generators
* Homogenized -- homogenize with respect to an indeterminate
* Interpolate -- interpolating polynomial
* Inverse -- multiplicative inverse
* IsHomog -- test whether given polynomials are homogeneous
* IsIn -- check if one object is contained in another
* IsTerm -- checks if the argument is a term
* IsZero -- test whether an object is zero
* Jacobian -- the Jacobian of a list of polynomials
* LC -- the leading coefficient of a polynomial or vector
* Len -- the length of an object
* LM -- the leading monomial of a polynomial or vector
* Log -- the list of exponents of the leading term of a polynomial
* LogToTerm -- returns a monomial (power-product) with given exponents
* LPP -- the leading power-product of a polynomial or vector
* LT -- the leading term of an object
* MapDown -- convert a constant polynomial to a number
* MDeg -- multi-degree of an polynomial
* Monic -- divide polynomials by their leading coefficients
* Monomials -- the list of monomials of a polynomial or vector
* NF -- normal form
* NFsAreZero -- test if normal forms are zero
* NR -- normal reduction
* Num, Den -- numerator, denominator
* Poly -- convert an expression into a polynomial
* Product, Sum -- the product or sum of the elements of a list
* QZP, ZPQ -- change field for polynomials and ideals
* Randomize, Randomized -- randomize the coefficients of a given polynomial
* Resultant -- the resultant of two polynomials
* Size -- the amount of memory used by an object
* StarPrint -- print polynomial with *'s for multiplications
* Subst -- substitute values for indeterminates
* Support -- the list of terms of a polynomial or vector
* Sylvester -- the Sylvester matrix of two polynomials
* WLog -- weighted list of exponents
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.10 Rational Functions
3.10.1 Introduction to Rational Functions
An object of type RATFUN in CoCoA represents a rational function,
i.e., a quotient of polynomials. Each rational function is
represented as P/Q where P and Q are polynomials (of type POLY) and
deg(Q) > 0. Common factors of the numerator and denominator are
automatically simplified. At present, rational functions in CoCoA are
only available over a field.
>EXAMPLE<
Use R ::= Q[x,y];
F := x/(x+y); -- a rational function
F*(x+y);
x
-------------------------------
(x^2-y^2)/(x+y); -- the result here is a polynomial
x - y
-------------------------------
The following algebraic operations on rational functions are supported:
F^N, +F, -F, F*G, F/G if G divides F, F+G, F-G,
where F, G are rational functions and N is an integer.
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.10.2 Numerators and Denominators for Rational Functions
If F is a variable holding a rational function, then F.Num and F.Den are the numerator and denominator, respectively. The functions 'Num' and 'Den', respectively, return the same. >EXAMPLE< F := x/(x+y); F.Num; x ------------------------------- F.Den; x+y ------------------------------- Den(F); x + y ------------------------------- See also: Den Num
3.10.3 Commands and Functions for Rational Functions
The following are commands and functions for rational functions:
* BringIn -- bring in objects from another ring
* Der -- the derivative of a rational function
* Fraction -- returns the quotient of its arguments
* Inverse -- multiplicative inverse
* IsZero -- test whether an object is zero
* Len -- the length of an object
* Num, Den -- numerator, denominator
* Product, Sum -- the product or sum of the elements of a list
* Size -- the amount of memory used by an object
* Subst -- substitute values for indeterminates
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.11 Ideals
3.11.1 Introduction to Ideals
An object of type IDEAL in CoCoA represents an ideal. An ideal is formed using using the command Ideal(P_1,...,P_n) where the P_i are generators for the ideal. >EXAMPLE< Use R ::= Q[x,y]; I := Ideal(x,y^2,2+xy^2); The following algebraic operations yield ideals: I^N, I+J, E*F, P:Q where: I and J are ideals; N is a non-negative integer; E,F are either both ideals or one is an ideal and the other is a polynomial; and the pair (P,Q) has the form (IDEAL,POLY), (IDEAL,IDEAL), (MODULE,VECTOR),(MODULE,MODULE). >EXAMPLE< Use R ::= Q[x,y]; I := Ideal(x,y^2,2+xy^2); I^2; Ideal(x^2, xy^2, x^2y^2 + 2x, y^4, xy^4 + 2y^2, x^2y^4 + 4xy^2 + 4) ------------------------------- J := Ideal(y); I+J; Ideal(x, y^2, xy^2 + 2, y) ------------------------------- I*J; Ideal(xy, y^3, xy^3 + 2y) -------------------------------
3.11.2 Commands and Functions for Ideals
The following are commands and functions for ideals:
* Colon, :, HColon -- ideal or module quotient
* Elim -- eliminate variables
* EquiIsoDec -- equidimensional isoradical decomposition
* GB.GetBettiMatrix -- returns the Betti matrix computed so far
* GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far
* GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far
* GB.GetRes -- returns the resolution computed so far
* GB.GetResLen -- returns the length of the resolution computed so far
* GB.ResReport -- status of an interactive resolution calculation
* GBasis -- calculate a Groebner basis
* GBM, HGBM -- intersection of ideals for zero-dimensional schemes
* GenRepr -- representation in terms of generators
* Gens -- list of generators of an ideal
* HIntersection, HIntersectionList -- intersection of ideals
* Homogenized -- homogenize with respect to an indeterminate
* Ideal -- convert an expression into an ideal
* IdealAndSeparatorsOfPoints -- ideal & separators for affine points
* IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
* IdealOfPoints -- ideal of a set of affine points
* IdealOfProjectivePoints -- ideal of a set of projective points
* Interreduce, Interreduced -- interreduce a list of polynomials or vectors
* Intersection, IntersectionList -- intersect lists, ideals, or modules
* IsIn -- check if one object is contained in another
* IsZero -- test whether an object is zero
* Len -- the length of an object
* LT -- the leading term of an object
* Max, Min -- a maximum or minimum element of a sequence or list
* MinGens -- list minimal generators
* Minimalize, Minimalized -- remove redundant generators
* MinSyzMinGens -- minimal generators of syzygies of minimal generators
* MonsInIdeal -- ideal generated by the monomials in an ideal
* NF -- normal form
* NFsAreZero -- test if normal forms are zero
* Product, Sum -- the product or sum of the elements of a list
* QuotientBasis -- vector space basis for zero-dimensional quotient rings
* QZP, ZPQ -- change field for polynomials and ideals
* Radical -- radical of an ideal
* RadicalOfUnmixed -- radical of an unmixed ideal
* ReducedGBasis -- compute a reduced Groebner basis
* Res -- free resolution
* Saturation, HSaturation -- saturation of ideals
* Size -- the amount of memory used by an object
* Syz -- syzygy modules
* SyzMinGens -- syzygy module for a set of minimal generators
* SyzOfGens -- syzygy module for a given set of generators
* Toric -- saturate toric ideals
* Toric.CheckInput -- check input to 'Toric'
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.12 Modules
3.12.1 Introduction to Modules
An object of type MODULE in CoCoA represents a submodule of a free
module. A module is represented by its generators as:
Module(V_1,...,V_n)
Each V_i has the form [P_1,...P_r], where r is the rank of the free
module containing the given module and each P_j is of type POLY.
As with ideals, information about a module can be accessed using the
same syntax as for records.
CoCoA supports quotient modules and modules, as described in the next
section. Shifts have been disabled in CoCoA 4.2.
>EXAMPLE<
Use S ::= Q[x,y];
M := Module([x,y^2,2+x^2y],[x,0,y]); -- define the submodule of S^3
-- generated by (x,y^2,2+x^2y) and (x,0,y)
GBasis(M);
[Vector(x, 0, y), Vector(x, y^2, x^2y + 2)]
-------------------------------
Describe M;
Record[Type = MODULE, Value = Record[Gens = [[x, y^2, x^2y + 2], [x,
0, y]], MRC = 1, GBasis = [[x, 0, y], [x, y^2, x^2y + 2]]]]
-------------------------------
M.GBasis;
[Vector(x, 0, y), Vector(x, y^2, x^2y + 2)]
-------------------------------
M.Gens[1];
Vector(x, y^2, x^2y + 2)
-------------------------------
M.NumComps; -- M is a submodule of a free module of rank 3
3
-------------------------------
3.12.2 Quotient Modules
If M is a CoCoA module which is a submodule of the free module R^k
for some ring R, then R^k/M represents a quotient module. It has type
TAGGED("Quotient").
>EXAMPLE<
Use R ::= Q[x,y];
M:= Module([x-y,x^2-y^2,x^3+xy^2],[y,x^2,x^2y]);
Q := R^3/M;
3.12.3 Shifts
===========================================
THIS FACILITY IS DISABLED IN COCOA 4
(See the new function 'GB.GetNthSyzShifts';
read below for an explanation of shifts.)
===========================================
One creates a shifted module in CoCoA using the 'Shifts' modifier:
Module(Shifts([P_1,...,P_r]),V_1,...,V_n)
where the P_i's are integers or monomials in the current ring and, as
usual, the V_i's are lists of polynomials, each with length r. This
object represents the submodule, generated by V_1,...,V_n, of the free
module of rank r which is the direct sum of R(P_1),...,R(P_r). Here,
R(P_i) is the ring R with shifted degrees. To explain these shifts,
recall that in a CoCoA ring, the weights of the indeterminates are
given by a weights matrix, say W. The (multi)weight of the i-th
indeterminate is given by the i-th column of W. By default, the
weights matrix is a single row of 1s. If P_i is an integer, then the
homogeneous part of R(P_i) of degree d is the homogeneous part of R of
degree d+P_i. If P_i is a monomial, then the homogeneous part of
R(P_i) in multidegree d is the homogeneous part of R in multidegree
d+deg_W(P_i).
>EXAMPLE<
Use R ::= Q[x,y,z];
M := Module([x,y,z],[x^2,y,0]);
LT(M.Gens[1]);
Vector(x, 0, 0)
-------------------------------
Deg(M.Gens[1]);
1
-------------------------------
Ss := Shifts([-3,-5,-2]);
M := Module(Ss,[x,y,z],[x^2,y,0]);
M;
Module(Shifts([-3, -5, -2]), [x, y, z], [x^2, y, 0])
-------------------------------
LT(M.Gens[1]); -- the leading term changes in the shifted module
Vector(0, y, 0)
-------------------------------
Deg(M.Gens[1]);
6
-------------------------------
Use S ::= Q[x,y,z],Weights(Mat([[1,2,3],[4,5,6]]));
M := Module(Shifts([xy, xz]), [x, y], [x,z]);
LT(M.Gens[1]);
Vector(0, y)
-------------------------------
MDeg(M.Gens[1]); -- multidegree of y in R(xz), i.e. of (xz)y in R
[6, 15]
-------------------------------
3.12.4 Commands and Functions for Modules
The following are commands and functions for modules:
* Colon, :, HColon -- ideal or module quotient
* E_ -- canonical vector
* Elim -- eliminate variables
* GB.GetBettiMatrix -- returns the Betti matrix computed so far
* GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far
* GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far
* GB.GetRes -- returns the resolution computed so far
* GB.GetResLen -- returns the length of the resolution computed so far
* GB.ResReport -- status of an interactive resolution calculation
* GBasis -- calculate a Groebner basis
* GenRepr -- representation in terms of generators
* Gens -- list of generators of an ideal
* Interreduce, Interreduced -- interreduce a list of polynomials or vectors
* Intersection, IntersectionList -- intersect lists, ideals, or modules
* IsIn -- check if one object is contained in another
* IsZero -- test whether an object is zero
* Len -- the length of an object
* LT -- the leading term of an object
* Max, Min -- a maximum or minimum element of a sequence or list
* MinGens -- list minimal generators
* Minimalize, Minimalized -- remove redundant generators
* Module -- convert an expression into a module
* NF -- normal form
* NFsAreZero -- test if normal forms are zero
* Rank -- rank of a module
* ReducedGBasis -- compute a reduced Groebner basis
* Res -- free resolution
* Size -- the amount of memory used by an object
* Syz -- syzygy modules
* SyzOfGens -- syzygy module for a given set of generators
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
3.13 Groebner Bases and Related Computations
- Introduction to Groebner Bases in CoCoA
- The Interactive Groebner Framework
- Hilbert-Driven Computations
3.13.1 Introduction to Groebner Bases in CoCoA
The heart of the CoCoA system is a implementation of Buchberger's algorithm for computing Groebner bases for ideals an modules over polynomial rings with coefficients in a field. CoCoA's Groebner basis engine can be used to compute Groebner bases, syzygies, free resolutions, Hilbert functions and Poincare series, and to eliminate variables and find minimal sets of generators. Considerable control over the computations is provided through CoCoA's "Interactive Groebner Framework." Groebner bases can be calculated over Q, but large calculations depending on Groebner bases will take much less time over finite fields. A common tactic is to work mod large primes to get an idea of behavior expected over Q. It would eventually be nice to have descriptions within this online help system of the specific algorithms use by CoCoA. For now, see 'Pointers to the Literature' for references.
3.13.2 Commands and Functions for Groebner-Type Computations
The following are the commands and functions for computations based
on Groebner bases. In addition to these, there are many commands that
provide finer control over the computations (see the next section:
'The Interactive Groebner Framework').
* Colon, :, HColon -- ideal or module quotient
* Dim -- the dimension of a ring or quotient object
* Elim -- eliminate variables
* EquiIsoDec -- equidimensional isoradical decomposition
* GB.Complete -- Complete an interactive Groebner-type calculation
* GB.GetBettiMatrix -- returns the Betti matrix computed so far
* GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far
* GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far
* GB.GetRes -- returns the resolution computed so far
* GB.GetResLen -- returns the length of the resolution computed so far
* GB.ResReport -- status of an interactive resolution calculation
* GB.Start_GBasis -- start interactive Groebner basis computation
* GB.Start_MinGens -- start interactive minimal generator calculation
* GB.Start_MinSyzMinGens -- start interactive calc., min. syzs. of min. gens.
* GB.Start_Res -- start interactive resolution computation
* GB.Start_Syz -- start interactive syzygy computation
* GB.Start_SyzMinGens -- start interactive calc. of syzygies of min. gens.
* GB.Stats -- status of an interactive Groebner-type calculation
* GB.Step, GB.Steps -- take steps in an interactive Groebner-type calculation
* GBasis -- calculate a Groebner basis
* GBM, HGBM -- intersection of ideals for zero-dimensional schemes
* GCD, LCM -- greatest common divisor, least common multiple
* Hilbert -- the Hilbert function
* HilbertPoly -- the Hilbert polynomial
* HIntersection, HIntersectionList -- intersection of ideals
* Homogenized -- homogenize with respect to an indeterminate
* HVector -- the h-vector of a ring or quotient object
* IdealAndSeparatorsOfPoints -- ideal & separators for affine points
* IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
* IdealOfPoints -- ideal of a set of affine points
* IdealOfProjectivePoints -- ideal of a set of projective points
* Intersection, IntersectionList -- intersect lists, ideals, or modules
* MinGens -- list minimal generators
* MinSyzMinGens -- minimal generators of syzygies of minimal generators
* Multiplicity -- the multiplicity (degree) of a ring or quotient object
* NF -- normal form
* NFsAreZero -- test if normal forms are zero
* Poincare, HilbertSeries -- the Poincare series
* Radical -- radical of an ideal
* RadicalOfUnmixed -- radical of an unmixed ideal
* ReducedGBasis -- compute a reduced Groebner basis
* Res -- free resolution
* Saturation, HSaturation -- saturation of ideals
* Syz -- syzygy modules
* SyzMinGens -- syzygy module for a set of minimal generators
* SyzOfGens -- syzygy module for a given set of generators
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
See also:
The Interactive Groebner Framework
3.13.3 The Interactive Groebner Framework
For the following computations: * Groebner bases * minimal generators * syzygies * free resolutions * elimination of variables CoCoA provides the following features: * step-by-step computation * monitoring of the execution (verbose mode) * various types of truncation (degree, resolution length, or regularity) * customization of algorithms (through the GROEBNER panel and P-Series). It works like this: instead of using one of the normal Groebner basis-type commands (listed in the previous section), start the computation with one of the commands, * GB.Start_GBasis -- start interactive Groebner basis computation * GB.Start_MinGens -- start interactive minimal generator calculation * GB.Start_Res -- start interactive resolution computation * GB.Start_Syz -- start interactive syzygy computation After starting the computation, the following commands are available: * GB.Complete -- Complete an interactive Groebner-type calculation * GB.GetBettiMatrix -- returns the Betti matrix computed so far * GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far * GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far * GB.GetRes -- returns the resolution computed so far * GB.GetResLen -- returns the length of the resolution computed so far * GB.ResReport -- status of an interactive resolution calculation * GB.Stats -- status of an interactive Groebner-type calculation * GB.Step, GB.Steps -- take steps in an interactive Groebner-type calculation * ReducedGBasis -- compute a reduced Groebner basis Almost all of these functions report more information if you set the Verbose flag in the GROEBNER panel by typing Set Verbose; (to unset, enter 'Unset Verbose'). For more possibilities, see 'Options in the GROEBNER Panel'. Use of the Interactive Groebner Framework is illustrated in the examples below.
3.13.4 Example: Interactive Groebner Basis Computation
>EXAMPLE< Use R ::= Q[t,x,y,z]; I := Ideal(t^3-x,t^4-y,t^5-z); GB.Start_GBasis(I); -- start the interactive framework I.GBasis; -- the Groebner basis is initially empty Null ------------------------------- GB.Step(I); -- a single step of the computation I.GBasis; I.GBasis; [t^3 - x] ------------------------------- GB.Steps(I,4); -- 4 more steps I.GBasis; [t^3 - x, -tx + y, t^2y - x^2] ------------------------------- GB.Complete(I); -- complete the computation I.GBasis; [t^3 - x, -tx + y, -ty + z, -y^2 + xz, -x^2 + tz, t^2z - xy] ------------------------------- ReducedGBasis(I); [t^3 - x, tx - y, ty - z, y^2 - xz, x^2 - tz, t^2z - xy] ------------------------------- Note that Groebner bases calculated in the interactive framework may not be reduced, as illustrated in the final step of the example.
3.13.5 Example: Verbose Mode
The following example illustrates the use of 'Verbose' mode. For more information, see 'Verbose'. >EXAMPLE< Set Verbose; Use R ::= Q[t,x,y,z]; I := Ideal(t^3-x,t^4-y,t^5-z); G := GBasis(I); ..................18 ------------------------------------ IPs IVs Gens GBases MinGens MinDeg ------------------------------------ 0 0 3 6 0 -1 ------------------------------------ Betti numbers: 18 steps of computation ------------------------------- Describe I; -- more information, (it would help to 'Set Indentation') Record[Type = IDEAL, Value = Record[Gens = [t^3 - x, t^4 - y, t^5 - z], GBasis = [t^3 - x, -tx + y, -ty + z, -y^2 + xz, -x^2 + tz, t^2z - xy], IVs = [ ], Rules = [t^3 - x, t^4 - y, t^5 - z, -tx + y, t^2y - x^2, x^3 - ty^2, -ty + z, -y^2 + xz, -x^2 + tz, t^2z - xy], Discrepancy = -1, KFL = [1, 7, 14, 4, 16, 9, 10, 20, 22, 23, 28, 18, 0, 39, 42]]] ------------------------------- See also: Describe Verbose
3.13.6 Example: Interactive Resolution Computation
In this example we compute the minimal free resolution of the ideal I
generated by the 2 by 2 minors of a catalecticant matrix, A, using the
interactive environment of the system. We define the ideal I, and
start the computation of its minimal free resolution using the
Hilbert-driven algorithm described in
A. Capani, G. De Dominicis, G. Niesi, L. Robbiano,
"Computing Minimal Finite Free Resolutions",
J. Pure Appl. Algebra, Vol. 117--118, Pages 105--117, 1997.
>EXAMPLE<
Use R ::= Z/(32003)[z[0..3,0..3,0..3]]; -- set up the ring
A := Mat([ -- define the ideal
[z[3,0,0], z[2,1,0], z[2,0,1]],
[z[2,1,0], z[1,2,0], z[1,1,1]],
[z[2,0,1], z[1,1,1], z[1,0,2]],
[z[1,2,0], z[0,3,0], z[0,2,1]],
[z[1,1,1], z[0,2,1], z[0,1,2]],
[z[1,0,2], z[0,1,2], z[0,0,3]]
]);
I := Ideal(Minors(2,A));
GB.Start_Res(I); -- start interactive framework
GB.Steps(I,1000); -- first 1000 steps
GB.GetRes(I);
0 --> R^176(-5) --> R^189(-4) --> R^105(-3) --> R^27(-2)
-------------------------------
GB.ResReport(I);
--------------------------------------------------------------
Minimal Pairs, : 650
Groebner Pairs : 14
Minimal (Type S) : 636
H-Killed (Type S0) : 9
--------------------------------------------------------------
-------------------------------
GB.Complete(I); -- complete the calculation
GB.GetRes(I);
0 --> R(-9) --> R^27(-7) --> R^105(-6) --> R^189(-5) -->
R^189(-4) --> R^105(-3) --> R^27(-2)
-------------------------------
GB.ResReport(I);
--------------------------------------------------------------
Minimal Pairs, : 730
Groebner Pairs : 25
Minimal (Type S) : 705
Minimal (Type Smin) : 616
Minimal (Type S0) : 89
H-Killed (Type S0) : 78
Hard (Type S0) : 11
--------------------------------------------------------------
-------------------------------
3.13.7 Example: Truncations
The user may assign one or more of three different truncation conditions to a module: DegTrunc, ResTrunc and RegTrunc; in this case the execution will stop when a bound is reached (see the examples, below). DEGREE TRUNCATION: >EXAMPLE< Use R ::= Z/(32003)[a,b,c,d,e]; I := Ideal(a+b+c+d, ab+bc+cd+da, abc+bcd+cda, abcd-e^4); I.DegTrunc := 3; GB.Start_GBasis(I); GB.Complete(I); LT(I.GBasis); [a, b^2, bc^2] ------------------------------- I.DegTrunc := 6; GB.Complete(I); LT(I.GBasis); [a, b^2, bc^2, bcd^2, c^2d^2, cd^4, be^4, d^2e^4] ------------------------------- RESOLUTION TRUNCATION: >EXAMPLE< Use R ::= Z/(32003)[x[1..10]]; I := Ideal(Indets()); I.ResTrunc := 4; GB.Start_Res(I); GB.Complete(I); GB.GetRes(I); 0 --> R^252(-5) --> R^210(-4) --> R^120(-3) --> R^45(-2) --> R^10(-1) ------------------------------- REGULARITY TRUNCATION: We know that the Castelnuovo regularity of I in the following example is 6. >EXAMPLE< Set Verbose; Use R_Gen ::= Z/(5)[x,y,z,t]; M := 3; N := 4; D := DensePoly(2); P := Mat([ [ Randomized(D) | J In 1..N ] | I In 1.. M]); I := Ideal(Minors(2,P)); GB.Start_Res(I); GB.Complete(I); -- text suppressed -- Betti numbers: 17 48 48 18 318 steps of computation I := Ideal(Minors(2,P)); GB.Start_Res(I); I.RegTrunc := 6; -- here we store the Castelnuovo Regularity GB.Complete(I); ... Betti numbers: 17 48 48 18 281 steps of computation GB.GetBettiMatrix(I); ------------------- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 16 0 0 0 32 0 0 48 0 0 17 0 0 0 -------------------
3.13.8 Hilbert-Driven Computations
CoCoA Groebner basis algorithms will use knowledge of a Poincare series to improve efficiency. >EXAMPLE< Use R ::= Q[w,x,y,z]; PS := HP.PSeries(1 + 2w + 3w^2 + 4w^3 + 5w^4 + 6w^5 + 5w^6 + 4w^7 + 3w^8 + 2w^9 + w^10,2); PS; (1 + 2w + 3w^2 + 4w^3 + 5w^4 + 6w^5 + 5w^6 + 4w^7 + 3w^8 + 2w^9 + w^10) / (1-w)^2 ------------------------------- I := Ideal((xy-zw)^3,(xz-yw)^3); I.PSeries := PS; -- this is how to let CoCoA know about the Poincare series G := GBasis(I);
Working the System
4.1 CoCoA Panels
4.1.1 Introduction to Panels
The user can customize some features of CoCoA by setting (or unsetting) some boolean options (ON/OFF) which are grouped in three panels. The function 'Panels' returns the list of panel names, and the settings in a particular panel can be seen using the function 'Panel' as illustrated in the example below. >EXAMPLE< Panels(); ["GENERAL", "GROEBNER"] ------------------------------- Panel(GENERAL); Echo............... : FALSE Timer.............. : FALSE Trace.............. : FALSE Indentation........ : FALSE TraceSources....... : FALSE SuppressWarnings... : FALSE ComputationStack... : FALSE -------------------------------
4.1.2 Setting Options
Each option name is unique, not just among the options in a particular panel, but among the options from all panels. The command 'Set' followed by an option name (without parentheses or quotes) sets the corresponding option to TRUE. Similarly, 'Unset' sets the option to FALSE. In addition, one may use the 'Set' command to set an option either true or false using the syntax: 'Set option-name := TRUE' or 'Set option-name := TRUE'. >EXAMPLE< Use R ::= Q[x,y,z]; L := [(x+y)^N | N In 1..3]; Set Indentation; -- print each component on a separate line L; L; [ x + y, x^2 + 2xy + y^2, x^3 + 3x^2y + 3xy^2 + y^3] ------------------------------- Unset Indentation; The function 'Option' takes as parameter an option name and returns the (boolean) value of the option. It is particularly useful within user-defined functions as illustrated in the example below: >EXAMPLE< Define Print_UnIndented(X) Opt := Option(Indentation); Unset Indentation; Print(X); Set Indentation := Opt; EndDefine; R := Record[X='test',L=[1,2,3]]; Set Indentation; Print_UnIndented(R); Record[L = [1, 2, 3], X = "test"] ------------------------------- UnSet Indentation; Print_UnIndented(R); Record[L = [1, 2, 3], X = "test"] -------------------------------
4.1.3 Options in the GENERAL Panel
The options in the GENERAL panel and their default settings are as follows: Panel(GENERAL); Echo............... : FALSE Timer.............. : FALSE Trace.............. : FALSE Indentation........ : FALSE TraceSources....... : FALSE SuppressWarnings... : FALSE ComputationStack... : FALSE ------------------------------- They are discussed separately, below.
4.1.4 Echo
If the Echo option is on, then the system echoes every command at the top level. When logging a CoCoA session, one would set Echo to TRUE in order to log both the input as well as the output in a CoCoA session (see 'OpenLog'). >EXAMPLE< 1+1; 2 ------------------------------- Set Echo; 1+1; 1 + 1 2 ------------------------------- L := [1,2,3]; L := [1, 2, 3] Unset Echo; SET(Echo, FALSE) L := [4,5,6]; -------------------------------
4.1.5 Timer
If the Timer option is on, then the system displays the execution time of each command submitted at top-level. >EXAMPLE< Use R ::= Z/(32003)[t,x,y,z],Lex; N:=31; I := Ideal(t^N+t^6+t-x, t^5-t-y, t^9-t-z); Set Timer; Null ------------------------------- T := GBasis(I); Cpu time = 2.88, User time = 30 ------------------------------- To time a single command, use 'Time'. For example, above, we could have written 'Time T := GBasis(I)' instead of setting the timer.
4.1.6 Trace
If the Trace option is on, then the system echoes every command at every level. This is useful for debugging programs. The following (toy) user-defined function returns the same error message for N = 3 and N = 6. Turning the Trace option on makes the sources of the trouble clear. >EXAMPLE< Define T(N) M := 1/(N-3); If N = 6 Then N := 3 EndIf; M := 1/(N-3); Return M; EndDefine; T(3); ------------------------------- ERROR: Division by zero CONTEXT: 1 / (N - 3) ------------------------------- T(6); ------------------------------- ERROR: Division by zero CONTEXT: 1 / (N - 3) ------------------------------- Set Trace; T(3); T(3) M := 1 / (N - 3); IF N = 6 THEN N := 3 END; M := 1 / (N - 3); Return(M); M := 1 / (N - 3) ------------------------------- ERROR: Division by zero CONTEXT: 1 / (N - 3) ------------------------------- T(6); T(6) M := 1 / (N - 3); IF N = 6 THEN N := 3 END; M := 1 / (N - 3); Return(M); M := 1 / (N - 3) IF N = 6 THEN N := 3 END N := 3 M := 1 / (N - 3) ------------------------------- ERROR: Division by zero CONTEXT: 1 / (N - 3) ------------------------------- See also: ComputationStack
4.1.7 Indentation
If the Indentation option is on, then the system performs some indentation on outputs. For example, each entry of a matrix will be printed on a new line. >EXAMPLE< Use R ::= Q[xyz] T := GBasis(Ideal(x^2-xy+z^3,xz-y^4,x^3+y^3+xyz+z^5)); T; [z^3 + x^2 - xy, -y^4 + xz, x^2z^2 - xyz^2 - x^3 - y^3 - xyz, x^4 - 2x^3y + x^2y^2 + x^3z + y^3z + xyz^2] ------------------------------- Set Indentation; T; [ z^3 + x^2 - xy, -y^4 + xz, x^2z^2 - xyz^2 - x^3 - y^3 - xyz, x^4 - 2x^3y + x^2y^2 + x^3z + y^3z + xyz^2] -------------------------------
4.1.8 TraceSources
If the TraceSources option is on, then the name of every file read with the 'Source' command will be echoed.
4.1.9 SuppressWarnings
If the SuppressWarnings option is on, then the system suppress warning statements. >EXAMPLE< Use Z/(4); -- WARNING: Coeffs are not in a field -- GBasis-related computations could fail to terminate or be wrong ------------------------------- ------------------------------- Set SuppressWarnings; Use Z/(4);
4.1.10 ComputationStack
If the ComputationStack option is on, a special variable named 'ComputationStack' contains a list tracing errors that occur during the execution of CoCoA commands. This option is useful for debugging programs. >EXAMPLE< Define Test(X) If X>=0 Then PrintLn(1/X) EndIf; EndDefine; Set ComputationStack; Test(0); ------------------------------- ERROR: Division by zero CONTEXT: 1 / X ------------------------------- S := ComputationStack; -- to save typing later S[1]; -- the command that produced the error PrintLn(1 / X) ------------------------------- S[2]; -- S[1] was part of an If-statement IF X >= 0 THEN PrintLn(1 / X) END ------------------------------- S[3]; -- the command issued by the user IF X >= 0 THEN PrintLn(1 / X) END; -------------------------------
4.1.11 Options in the GROEBNER Panel
The options in the GROEBNER panel and their default settings are as follows: Panel(GROEBNER); Sugar........... : TRUE FullRed......... : TRUE SingleStepRed... : FALSE Verbose......... : FALSE ------------------------------- They are discussed separately, below.
4.1.12 Sugar
If the Sugar option is on, as it is by default, then the critical pairs are processed by using the "sugar" strategy: the pairs are processed in an order which is as close as possible to the order which would have been chosen if the polynomials had been homogeneous. For details, see the article: A. Giovini, T. Mora, G. Niesi, L. Robbiano, C. Traverso, "`One sugar cube, please' or selection strategies in the Buchberger algorithm," In Proc. ISSAC '91, 49--54 (1991), Stephen M. Watt, editor, New York, ACM Press.
4.1.13 FullRed
If FullRed is set to TRUE, then when a normal form is required in any Groebner-type computation, CoCoA will reduce all monomials in a polynomial; if FullRed is FALSE, only the leading terms will be reduced. The default is to have FullRed set to TRUE. >EXAMPLE< UnSet FullRed; Use R ::= Q[x,y]; Interreduced([xy^3+y^2+x,x]); [x, y^2 + x] ------------------------------- Set FullRed; Interreduced([xy^3+y^2+x,x]); [x, y^2] ------------------------------- See also: Interreduced
4.1.14 SingleStepRed
Sorry: documentation not yet available.
4.1.15 Verbose
Sorry: documentation not yet available.
4.1.16 Commands and Functions for Panels
The following are commands and functions for panels:
* Option -- status of a panel option
* Panel -- print status of a panel's options
* Panels -- list of CoCoA panels
* Reset -- reset panels and random number seed to defaults
* ResetPanels -- reset panels to their default values
* Set, Unset -- set and unset panel options
Details can be found by looking up each item by name. Online, try
'?ItemName' or 'H.Syntax("ItemName")'.
4.2 CoCoA's Help System
4.2.1 Online Help
CoCoA's online help is roughly divided into two parts: a manual and a
list of commands. The manual includes a tutorial which can be started
by entering
?tutorial
If you are a new user of CoCoA, the tutorial is a good place to start.
To see the titles of the sections of the online manual, enter
'H.Toc()'; to see a list of commands, enter 'H.Commands('')'.
Each section of the manual and each command is uniquely identified by
a set of keywords. The set of keywords always includes the title of
the section or the title of the command. The online help command '?'
takes a string from the user and searches for a match among the
keywords. For instance, '?gbasis' will display information
about the function 'GBasis'. Note that when using '?', the keyword
does not appear in quotes. Also, a semicolon is not required.
The command '??' will return a list of entries in the manual that
contain a given keyword. For instance, '??gbasis' returns
See:
GB.Start_GBasis
GBasis
ReducedGBasis
-------------------------------
Each command is associated with a list of topics. (This applies only
to commands, not to sections of the manual.) The online help function
'H.Commands' takes a string from the user and searches for all matches
among these topics. For each match, the title of the command and a
brief description is displayed. For instance, 'H.Commands('poly')'
will find all commands having to do with polynomials. Information
about a specific command can then be retrieved with '?'. A list of
topics is provided by 'H.Commands()', with no argument.
IMPORTANT NOTE: Searches are case insensitive and your keyword need
only be a substring to make a match.
For a thorough description of the search function '?', type '? ?' with
a space bewteen the two questions marks.
Tips on using online help and summary of the online help functions
appear below. Enter 'H.Browse();' to see the next section.
4.2.2 Quick Tips for Using Online Help
Here are some tips for using the online help system:
1. Searches are case insensitive and your search string need only be a
substring of a keyword to make a match. Thus, for instance, to find
the section of the manual entitled "Commands and Functions for
Polynomials", it is enough to type: '?for poly'.
2. In general, it is best to start with short keywords in order to
maximize the number of matches.
3. If you cannot find a match using '?', try using 'H.Command' to
search by type. For example, suppose you are looking for a command
that will give the remainder of the division of one polynomial into
another. Trying '?remainder' produces no matches. You know
that the command you are looking for operates on polynomials, so try
'H.Command('poly')'. It produces a list of a little over 30 commands,
among which will be listed:
* DivAlg -- division algorithm.
4. The command '??keyword', with two question marks, will list all
matches even if 'keyword' exactly matches a keyword for the online
help system.
4.2.3 Commands and Functions for Online Help
The following are commands and functions for CoCoA online help:
* H.Browse -- browse the online help system
* H.Commands -- list commands according to type
* H.Man -- search online help system (see 'Man')
* H.OutCommands -- print command descriptions to a file
* H.OutManual -- print the online manual to a file
* H.SetMore, H.UnSetMore -- more-device for online help
* H.Syntax -- display the syntax of a command
* H.Tips -- quick tips for using online help
* H.Toc -- display the Table Of Contents of the online manual
* H.Tutorial -- run the CoCoA tutorial
* ?, Man -- search online help system
Details can be found by looking up each item by name. Online, try
'?ItemName')' or 'H.Syntax('ItemName')'. A good place to start is
with the command, '?', itself. To see more information about '?',
enter '? ?' (with a space between the two question marks)
4.2.4 Other Help
1. A user may provide help and sometimes (rarely) get help for a
user-defined function using the 'Help' feature of the 'Define'
command.
>EXAMPLE<
Help();
Type Help < Op > to get help on operator < Op >
-------------------------------
Help('GBasis'); -- note the typical response, one of the main
-- motivations for the author of the online manual.
No help available for GBasis
-------------------------------
2. The command, 'Describe', can be used to find more information about
functions.
>EXAMPLE<
Describe Function('Insert');
DEFINE Insert(L,I,O)
$cocoa/list.Insert(L,I,O)
END
-------------------------------
Describe Function('$cocoa/list.Insert');
DEFINE Insert(L,I,O)
IF NOT(Type(L) = LIST) THEN
Return(Error(ERR.BAD_PARAMS,": expected LIST"))
ELSIF I = Len(L) + 1 THEN
Append(L,O)
ELSIF I > Len(L) THEN
Return(Error(ERR.INDEX_TOO_BIG,I))
ELSIF I <= 0 THEN
Return(Error(ERR.INDEX_NEG,I))
ELSE
L := Concat([L[J]|J IN 1..(I - 1)],[O],[L[J]|J IN I..Len(L)]);
END;
END
-------------------------------
3. The function, 'Functions', may be used to list all functions in a
package. Note: 'Functions('$cocoa/user')' lists all current
user-defined functions.
>EXAMPLE<
Functions('$cocoa/mat');
[About(), Man(), Identity(N), Transposed(M), Submat(M,Rows,Cols),
Jacobian(S), Resultant(F,G,X), DirectSum(M1,M2), BlockMatrix(LL),
ColumnVectors(M), Complement(M,I,J), Bip(M,J), Pfaffian(M),
Sylvester(F,G,X), ExtendRows(M,N), PushRows(M,N), ConcatRows(L),
PkgName()]
-------------------------------
The list of packages is given by 'Packages()'.
4. The function 'Starting(S)' where S is a string returns a list of all
functions starting with the string S.
>EXAMPLE<
Starting('Su');
["SubstPoly", "SubSet", "Submat", "Sum", "Subst", "Support"]
-------------------------------
4.3 Fine Tuning At Start-up
4.3.1 User Initialization
At the beginning of a CoCoA session, CoCoA reads in a file called 'init.coc'. This file performs certain initialization routines, reads in standard packages, and sets up global aliases for the packages. It also reads in a file called 'userinit.coc'. It is in this latter file that users should put their own commands to be run when CoCoA starts. For example, suppose a user wants a file called 'MyFile.coc'--- containing function definitions, variable assignments, etc.---to automatically be read into the CoCoA system on start-up. It suffices to add the following line to 'userinit.coc': <<'MyFile.coc'; To load the package with identifier $contrib/mypackage, contained in a file 'MyPackage.pkg' and use MP as an alias, it suffices to add the following lines to 'userinit.coc' (you may want to look at the section of the manual entitled "Package Sourcing" for more details): <<'MyPackage.pkg'; Alias MP := $contrib/mypackage;
4.4 CoCoA Interfaces
4.4.1 CoCoA on a Macintosh
The CoCoA user interface on the Macintosh is based on Mel Park's
PlainText (v.1.6) which handles very large text files (larger than
32K). It uses standard Macintosh editing techniques, so Macintosh
users should be familiar with its basic operations.
Double-clicking on the CoCoA icon or on the icon of a CoCoA document
will start up the system. The system draws the menu bar, opens a text
editing window and loads the CoCoA library and then possibly user's
libraries (via the 'userinit.coc' file).
After the system is started, it is ready to receive and execute
commands. To execute a CoCoA command, type it into the window, ending
it with a semicolon, then press the 'enter' key. If the command
occupies more than one line then highlight the whole command using the
mouse and then press the "enter" key.
At this point the following part of the text of the active window is
taken as being the 'current command':
* if there is no selection (the cursor is blinking somewhere), then
the row containing the cursor is taken as current command;
* if there is a non-null selection range, then the whole selection is
taken as current command (in this way the system can process multiline
commands).
The editor uses all the standard Macintosh editing techniques as well as
some special ones:
* Double-clicking on a word select the entire word.
* Triple-clicking anywhere in a line selects the whole line.
* Double-clicking on or just before a parenthesis, a bracket, or a
brace, i.e. one of following symbols '(' , ')' , '[' , ']' , '{' ,
'}' causes all the text between that symbol and its matching symbol
to become selected.
IMPORTANT NOTE. Devices of type FILE are not yet available with the
Macintosh interface.
4.4.2 CoCoA under Unix
Probably the best way to run CoCoA under Unix is through the editor, emacs, in shell-mode. In that way, one may easily edit or repeat commands. From within emacs, issue the command 'META-x shell'. You will be presented with a Unix prompt within emacs. Change to the CoCoA directory, and start CoCoA. For more information about shell-mode, issue the emacs command 'Control-h m' after the shell is started. (You may want to use the emacs command: 'META-x set-variable comint-scroll-show-maximum-output' to set the variable comint-scroll-show-maximum-output to the value 1.) If running CoCoA in an xterm, it may be best to first start the xterm with the command 'xterm -sb -sl 512' (scroll bar enabled, saving 512 lines). In addition, you may want to increase the vertical size of your window, e.g., 'xterm -sb -sl 512 -geometry 80x40'. In that way, output that scrolls off of the screen is captured and easily reviewed. At any rate, complicated CoCoA command sequences or any sequences that you may want to repeat should be saved in a text editor. The commands can then be executed by copying and pasting into a CoCoA window or using the 'Source' command. In addition, you may want to keep a log of your CoCoA session using the command 'OpenLog'.
4.4.3 CoCoA under Windows/DOS
Sorry this section is still under construction. Probably the best advice I could give is to repartition your disk and install Linux. ;>
Alphabetic List of Commands
- .. -- range operator
- >< -- Cartesian product
- Abs -- absolute value of a number
- Adjoint -- adjoint matrix
- Alias -- define aliases for package names
- Alias In -- temporarily override global aliases
- Aliases -- list of global aliases
- Ascii -- convert between characters and ascii code
- Append -- append an object to an existing list
- Bin -- binomial coefficient
- BinExp, EvalBinExp -- binomial expansion
- Block -- group several commands into a single command
- BlockMatrix -- create a block matrix
- Break -- break out of a loop
- BringIn -- bring in objects from another ring
- Call -- apply a function to given arguments
- Cast -- type conversion
- Catch -- catch an error
- Characteristic -- the characteristic of a ring
- Ciao -- quit CoCoA
- Clear -- clear the working memory or a ring-bound memory
- Close -- close a device
- CocoaLibrary -- returns the path to the CoCoA libraries
- CocoaLimits -- limits on exponents and ring characteristics
- Coefficients -- list of coefficients of a polynomial or vector
- CoeffOfTerm -- coefficient of a term of a polynomial or vector
- Colon, :, HColon -- ideal or module quotient
- ColumnVectors -- the list of column vectors of a matrix
- Comp -- the N-th component of a list
- Comps -- list of components of a vector
- Concat, ConcatLists -- concatenate lists or lists of lists, respectively
- Cond -- conditional expression
- Contrib -- list of contributors to CoCoA
- Count -- count the objects in a list
- CurrentRing -- the current ring
- Dashes -- returns string of dashes
- Date -- the date
- Define -- define a function
- Defined -- check if an expression has been defined
- Deg -- the degree of a polynomial or vector
- DegLexMat, DegRevLexMat, LexMat, XelMat -- matrices for std. term-orderings
- Delete -- delete variables from the working memory
- DensePoly -- the sum of all power-products of a given degree
- Der -- the derivative of a rational function
- Describe -- information about an expression
- Destroy -- delete rings
- Det -- the determinant of a matrix
- Diff -- returns the difference between two lists
- Dim -- the dimension of a ring or quotient object
- Discriminant -- the discriminant of a polynomial
- Distrib -- the distribution of objects in a list
- Div, Mod -- quotient and remainder for integers
- DivAlg -- division algorithm
- E_ -- canonical vector
- Elim -- eliminate variables
- EqSet -- checks if the set of elements in two lists are equal
- Equals -- returns a string of equal signs
- EquiIsoDec -- equidimensional isoradical decomposition
- Error -- return an error message
- Eval -- substitute numbers or polynomials for indeterminates
- EvalHilbertFn -- evaluate the Hilbert function
- Fact -- factorial function
- Factor -- factor a polynomial
- Fields -- list the fields of a record
- First -- the first N elements of a list
- FirstNonZero, FirstNonZeroPos -- the first non-zero entry in a vector
- Flatten -- flatten a list
- FloatStr, MantissaAndExponent -- convert rational number to a float string
- For -- loop command
- Foreach -- loop command
- Format -- convert object to formatted string
- Fraction -- returns the quotient of its arguments
- Function -- return a function
- Functions -- list the functions of a package
- GB.Complete -- Complete an interactive Groebner-type calculation
- GB.GetBettiMatrix -- returns the Betti matrix computed so far
- GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far
- GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far
- GB.GetRes -- returns the resolution computed so far
- GB.GetResLen -- returns the length of the resolution computed so far
- GB.ResReport -- status of an interactive resolution calculation
- GB.Start_GBasis -- start interactive Groebner basis computation
- GB.Start_MinGens -- start interactive minimal generator calculation
- GB.Start_MinSyzMinGens -- start interactive calc., min. syzs. of min. gens.
- GB.Start_Res -- start interactive resolution computation
- GB.Start_Syz -- start interactive syzygy computation
- GB.Start_SyzMinGens -- start interactive calc. of syzygies of min. gens.
- GB.Stats -- status of an interactive Groebner-type calculation
- GB.Step, GB.Steps -- take steps in an interactive Groebner-type calculation
- GBasis -- calculate a Groebner basis
- GBM, HGBM -- intersection of ideals for zero-dimensional schemes
- GCD, LCM -- greatest common divisor, least common multiple
- GenericPoints -- random projective points
- GenRepr -- representation in terms of generators
- Gens -- list of generators of an ideal
- Get -- read characters from a device
- GetErrMesg -- returns the message associated with an error
- GlobalMemory -- contents of global memory
- H.Browse -- browse the online help system
- H.Commands -- list commands according to type
- H.Man -- search online help system (see '?, Man')
- H.OutCommands -- print command descriptions to a file
- H.OutManual -- print the online manual to a file
- H.SetMore, H.UnSetMore -- more-device for online help
- H.Syntax -- display the syntax of a command
- H.Tips -- quick tips for using online help
- H.Toc -- display the Table Of Contents of the online manual
- H.Tutorial -- run the CoCoA tutorial
- Head -- the first element of a list
- Help -- extend online help for user-defined functions
- Hilbert -- the Hilbert function
- HilbertPoly -- the Hilbert polynomial
- HIntersection, HIntersectionList -- intersection of ideals
- Homogenized -- homogenize with respect to an indeterminate
- HVector -- the h-vector of a ring or quotient object
- Ideal -- convert an expression into an ideal
- IdealAndSeparatorsOfPoints -- ideal & separators for affine points
- IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
- IdealOfPoints -- ideal of a set of affine points
- IdealOfProjectivePoints -- ideal of a set of projective points
- Identity -- the identity matrix
- If -- conditional statement
- Image -- ring homomorphism
- In -- create a list satisfying given conditions
- Indet -- individual indeterminates
- IndetInd -- the index of an indeterminate
- IndetIndex -- index of an indeterminate
- Indets -- list of current indeterminates
- IndetName -- the name of an indeterminate
- Insert, Remove -- insert or remove an object in a list
- Interpolate -- interpolating polynomial
- Interreduce, Interreduced -- interreduce a list of polynomials or vectors
- Intersection, IntersectionList -- intersect lists, ideals, or modules
- Inverse -- multiplicative inverse
- IO.SprintTrunc -- convert to a string and truncate
- IsEven, IsOdd -- test whether an integer is even or odd
- IsHomog -- test whether given polynomials are homogeneous
- IsIn -- check if one object is contained in another
- IsNumber -- checks if the argument is a number
- IsPosTo, IsToPos -- checks the module term-ordering of a ring
- IsPrime -- prime integer test
- Isqrt -- computes the (truncated) square root of an integer
- IsTerm -- checks if the argument is a term
- IsZero -- test whether an object is zero
- Jacobian -- the Jacobian of a list of polynomials
- Last -- the last N elements of a list
- Latex -- LaTeX formatting
- LC -- the leading coefficient of a polynomial or vector
- Len -- the length of an object
- LinKer -- find the kernel of a matrix
- LinSol -- find a solution to a linear system
- List -- convert an expression into a list
- LM -- the leading monomial of a polynomial or vector
- Log -- the list of exponents of the leading term of a polynomial
- LogToTerm -- returns a monomial (power-product) with given exponents
- LPos -- the position of the leading power-product in a vector
- LPP -- the leading power-product of a polynomial or vector
- LT -- the leading term of an object
- ?, Man -- search online help system
- MapDown -- convert a constant polynomial to a number
- Mat -- convert an expression into a matrix
- Max, Min -- a maximum or minimum element of a sequence or list
- MDeg -- multi-degree of an polynomial
- Memory -- contents of local memory or ring-bound memory
- MinGens -- list minimal generators
- Minimalize, Minimalized -- remove redundant generators
- Minors -- list of minor determinants of a matrix
- MinSyzMinGens -- minimal generators of syzygies of minimal generators
- Mod2Rat -- reconstructing rationals from modular integers
- Module -- convert an expression into a module
- Monic -- divide polynomials by their leading coefficients
- Monomials -- the list of monomials of a polynomial or vector
- MonsInIdeal -- ideal generated by the monomials in an ideal
- More -- print a string, N lines at a time
- Multiplicity -- the multiplicity (degree) of a ring or quotient object
- NewId -- create a new identifier
- NewList -- create a new list
- NewMat -- create a new matrix
- NewVector -- create a new vector
- NextPrime -- find the next largest prime number
- NF -- normal form
- NFsAreZero -- test if normal forms are zero
- NonZero -- remove zeroes from a list
- Not, And, Or -- boolean operators
- NR -- normal reduction
- Num, Den -- numerator, denominator
- NumComps -- the number of components of a vector
- NumIndets -- number of indeterminates
- OpenIFile, OpenOFile -- open input or output file
- OpenIString, OpenOString -- open input or output string
- OpenLog, CloseLog -- open or close a log of a CoCoA session
- Option -- status of a panel option
- Ord -- matrix defining a term-ordering
- Packages -- list of loaded packages
- Panel -- print status of a panel's options
- Panels -- list of CoCoA panels
- Pfaffian -- the Pfaffian of a skew-symmetric matrix
- PkgName -- returns the name of a package
- Poincare, HilbertSeries -- the Poincare series
- Poly -- convert an expression into a polynomial
- Print On -- print to an output device
- Print, PrintLn -- print the value of an expression
- Product, Sum -- the product or sum of the elements of a list
- Quit -- quit CoCoA
- QuotientBasis -- vector space basis for zero-dimensional quotient rings
- QZP, ZPQ -- change field for polynomials and ideals
- Radical -- radical of an ideal
- RadicalOfUnmixed -- radical of an unmixed ideal
- Rand -- random integer
- Randomize, Randomized -- randomize the coefficients of a given polynomial
- Rank -- rank of a module
- Record -- create a record
- ReducedGBasis -- compute a reduced Groebner basis
- Repeat -- loop command
- Res -- free resolution
- Resultant -- the resultant of two polynomials
- Reset -- reset panels and random number seed to defaults
- ResetPanels -- reset panels to their default values
- Return -- exit from a structured command
- Reverse, Reversed -- reverse a list
- Ring -- returns the ring with a given name
- RingEnv -- name of the current ring
- RingEnvs -- names of all defined rings
- Saturation, HSaturation -- saturation of ideals
- ScalarProduct -- scalar product
- Seed -- seed for 'Rand'
- SeparatorsOfPoints -- separators for affine points
- SeparatorsOfProjectivePoints -- separators for projective points
- Set -- remove duplicates from a list
- Set, Unset -- set and unset panel options
- Shape -- extended list of types involved in an expression
- Size -- the amount of memory used by an object
- Skip -- does nothing
- Sort, Sorted -- sort a list
- SortBy, SortedBy -- sort a list
- Source, << -- read commands from a file or device
- Spaces -- return a string of spaces
- Sprint -- convert to a string
- StarPrint -- print polynomial with *'s for multiplications
- Starting -- list functions starting with a given string
- Submat -- submatrix
- SubSet -- checks if the elements of one list are a subset of another
- Subst -- substitute values for indeterminates
- Support -- the list of terms of a polynomial or vector
- Sylvester -- the Sylvester matrix of two polynomials
- Syz -- syzygy modules
- SyzMinGens -- syzygy module for a set of minimal generators
- SyzOfGens -- syzygy module for a given set of generators
- Tag -- returns the tag string of an object
- Tagged, Untagged, @ -- tag or untag an object for pretty printing
- Tail -- remove the first element of a list
- TensorMat -- returns the tensor product of two matrices
- Toric -- saturate toric ideals
- Toric.CheckInput -- check input to 'Toric'
- Transposed -- the transposition of a matrix
- Type -- the data type of an expression
- TypeOfCoeffs -- type of the coefficients of the current ring
- Types -- lists all data types
- Use -- command for making a ring active
- Using -- perform commands in non-active ring
- Var -- function calls by reference, other complex referencing
- Vector -- create a vector
- WeightsList -- first row of the weights matrix
- WeightsMatrix -- matrix of generalized weights for indeterminates
- While -- loop command
- WithoutNth -- removes the N-th component from a list
- WLog -- weighted list of exponents
List of Commands by Type
- boolean
- EqSet -- checks if the set of elements in two lists are equal
- IsEven, IsOdd -- test whether an integer is even or odd
- IsHomog -- test whether given polynomials are homogeneous
- IsIn -- check if one object is contained in another
- IsNumber -- checks if the argument is a number
- IsPosTo, IsToPos -- checks the module term-ordering of a ring
- IsZero -- test whether an object is zero
- Not, And, Or -- boolean operators
- SubSet -- checks if the elements of one list are a subset of another
- branching
- cast
- error
- function
- groebner
- Colon, :, HColon -- ideal or module quotient
- Dim -- the dimension of a ring or quotient object
- Elim -- eliminate variables
- EquiIsoDec -- equidimensional isoradical decomposition
- GB.Complete -- Complete an interactive Groebner-type calculation
- GB.GetBettiMatrix -- returns the Betti matrix computed so far
- GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far
- GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far
- GB.GetRes -- returns the resolution computed so far
- GB.GetResLen -- returns the length of the resolution computed so far
- GB.ResReport -- status of an interactive resolution calculation
- GB.Start_GBasis -- start interactive Groebner basis computation
- GB.Start_MinGens -- start interactive minimal generator calculation
- GB.Start_MinSyzMinGens -- start interactive calc., min. syzs. of min. gens.
- GB.Start_Res -- start interactive resolution computation
- GB.Start_Syz -- start interactive syzygy computation
- GB.Start_SyzMinGens -- start interactive calc. of syzygies of min. gens.
- GB.Stats -- status of an interactive Groebner-type calculation
- GB.Step, GB.Steps -- take steps in an interactive Groebner-type calculation
- GBasis -- calculate a Groebner basis
- GBM, HGBM -- intersection of ideals for zero-dimensional schemes
- GCD, LCM -- greatest common divisor, least common multiple
- Hilbert -- the Hilbert function
- HilbertPoly -- the Hilbert polynomial
- HIntersection, HIntersectionList -- intersection of ideals
- Homogenized -- homogenize with respect to an indeterminate
- HVector -- the h-vector of a ring or quotient object
- IdealAndSeparatorsOfPoints -- ideal & separators for affine points
- IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
- IdealOfPoints -- ideal of a set of affine points
- IdealOfProjectivePoints -- ideal of a set of projective points
- Intersection, IntersectionList -- intersect lists, ideals, or modules
- MinGens -- list minimal generators
- MinSyzMinGens -- minimal generators of syzygies of minimal generators
- Multiplicity -- the multiplicity (degree) of a ring or quotient object
- NF -- normal form
- NFsAreZero -- test if normal forms are zero
- Poincare, HilbertSeries -- the Poincare series
- Radical -- radical of an ideal
- RadicalOfUnmixed -- radical of an unmixed ideal
- ReducedGBasis -- compute a reduced Groebner basis
- Res -- free resolution
- Saturation, HSaturation -- saturation of ideals
- Syz -- syzygy modules
- SyzMinGens -- syzygy module for a set of minimal generators
- SyzOfGens -- syzygy module for a given set of generators
- groebner-basic
- Colon, :, HColon -- ideal or module quotient
- Dim -- the dimension of a ring or quotient object
- Elim -- eliminate variables
- GBasis -- calculate a Groebner basis
- GCD, LCM -- greatest common divisor, least common multiple
- Hilbert -- the Hilbert function
- HilbertPoly -- the Hilbert polynomial
- HIntersection, HIntersectionList -- intersection of ideals
- Homogenized -- homogenize with respect to an indeterminate
- HVector -- the h-vector of a ring or quotient object
- Intersection, IntersectionList -- intersect lists, ideals, or modules
- MinGens -- list minimal generators
- MinSyzMinGens -- minimal generators of syzygies of minimal generators
- Multiplicity -- the multiplicity (degree) of a ring or quotient object
- NF -- normal form
- NFsAreZero -- test if normal forms are zero
- Poincare, HilbertSeries -- the Poincare series
- Res -- free resolution
- Saturation, HSaturation -- saturation of ideals
- Syz -- syzygy modules
- SyzMinGens -- syzygy module for a set of minimal generators
- SyzOfGens -- syzygy module for a given set of generators
- groebner-interactive
- GB.Complete -- Complete an interactive Groebner-type calculation
- GB.GetBettiMatrix -- returns the Betti matrix computed so far
- GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far
- GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far
- GB.GetRes -- returns the resolution computed so far
- GB.GetResLen -- returns the length of the resolution computed so far
- GB.ResReport -- status of an interactive resolution calculation
- GB.Stats -- status of an interactive Groebner-type calculation
- GB.Step, GB.Steps -- take steps in an interactive Groebner-type calculation
- ReducedGBasis -- compute a reduced Groebner basis
- groebner-start
- GB.Start_GBasis -- start interactive Groebner basis computation
- GB.Start_MinGens -- start interactive minimal generator calculation
- GB.Start_MinSyzMinGens -- start interactive calc., min. syzs. of min. gens.
- GB.Start_Res -- start interactive resolution computation
- GB.Start_Syz -- start interactive syzygy computation
- GB.Start_SyzMinGens -- start interactive calc. of syzygies of min. gens.
- help
- Describe -- information about an expression
- Functions -- list the functions of a package
- H.Browse -- browse the online help system
- H.Commands -- list commands according to type
- H.Man -- search online help system (see '?, Man')
- H.OutCommands -- print command descriptions to a file
- H.OutManual -- print the online manual to a file
- H.SetMore, H.UnSetMore -- more-device for online help
- H.Syntax -- display the syntax of a command
- H.Toc -- display the Table Of Contents of the online manual
- H.Tutorial -- run the CoCoA tutorial
- Help -- extend online help for user-defined functions
- ?, Man -- search online help system
- Starting -- list functions starting with a given string
- hilbert
- Dim -- the dimension of a ring or quotient object
- EvalHilbertFn -- evaluate the Hilbert function
- Hilbert -- the Hilbert function
- HilbertPoly -- the Hilbert polynomial
- HVector -- the h-vector of a ring or quotient object
- Multiplicity -- the multiplicity (degree) of a ring or quotient object
- Poincare, HilbertSeries -- the Poincare series
- ideal
- Colon, :, HColon -- ideal or module quotient
- Elim -- eliminate variables
- EquiIsoDec -- equidimensional isoradical decomposition
- GB.GetBettiMatrix -- returns the Betti matrix computed so far
- GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far
- GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far
- GB.GetRes -- returns the resolution computed so far
- GB.GetResLen -- returns the length of the resolution computed so far
- GB.ResReport -- status of an interactive resolution calculation
- GBasis -- calculate a Groebner basis
- GBM, HGBM -- intersection of ideals for zero-dimensional schemes
- GenRepr -- representation in terms of generators
- Gens -- list of generators of an ideal
- HIntersection, HIntersectionList -- intersection of ideals
- Homogenized -- homogenize with respect to an indeterminate
- Ideal -- convert an expression into an ideal
- IdealAndSeparatorsOfPoints -- ideal & separators for affine points
- IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
- IdealOfPoints -- ideal of a set of affine points
- IdealOfProjectivePoints -- ideal of a set of projective points
- Interreduce, Interreduced -- interreduce a list of polynomials or vectors
- Intersection, IntersectionList -- intersect lists, ideals, or modules
- IsIn -- check if one object is contained in another
- IsZero -- test whether an object is zero
- Len -- the length of an object
- LT -- the leading term of an object
- Max, Min -- a maximum or minimum element of a sequence or list
- MinGens -- list minimal generators
- Minimalize, Minimalized -- remove redundant generators
- MinSyzMinGens -- minimal generators of syzygies of minimal generators
- MonsInIdeal -- ideal generated by the monomials in an ideal
- NF -- normal form
- NFsAreZero -- test if normal forms are zero
- Product, Sum -- the product or sum of the elements of a list
- QuotientBasis -- vector space basis for zero-dimensional quotient rings
- QZP, ZPQ -- change field for polynomials and ideals
- Radical -- radical of an ideal
- RadicalOfUnmixed -- radical of an unmixed ideal
- ReducedGBasis -- compute a reduced Groebner basis
- Res -- free resolution
- Saturation, HSaturation -- saturation of ideals
- Size -- the amount of memory used by an object
- Syz -- syzygy modules
- SyzMinGens -- syzygy module for a set of minimal generators
- SyzOfGens -- syzygy module for a given set of generators
- Toric -- saturate toric ideals
- Toric.CheckInput -- check input to 'Toric'
- integer
- Abs -- absolute value of a number
- Bin -- binomial coefficient
- BinExp, EvalBinExp -- binomial expansion
- Div, Mod -- quotient and remainder for integers
- EvalHilbertFn -- evaluate the Hilbert function
- Fact -- factorial function
- Fraction -- returns the quotient of its arguments
- GCD, LCM -- greatest common divisor, least common multiple
- GenericPoints -- random projective points
- Inverse -- multiplicative inverse
- IsEven, IsOdd -- test whether an integer is even or odd
- IsNumber -- checks if the argument is a number
- IsPrime -- prime integer test
- Isqrt -- computes the (truncated) square root of an integer
- IsZero -- test whether an object is zero
- Len -- the length of an object
- Max, Min -- a maximum or minimum element of a sequence or list
- Mod2Rat -- reconstructing rationals from modular integers
- NextPrime -- find the next largest prime number
- Num, Den -- numerator, denominator
- Product, Sum -- the product or sum of the elements of a list
- Rand -- random integer
- Seed -- seed for 'Rand'
- Size -- the amount of memory used by an object
- io
- Block -- group several commands into a single command
- Close -- close a device
- Format -- convert object to formatted string
- Get -- read characters from a device
- IO.SprintTrunc -- convert to a string and truncate
- Latex -- LaTeX formatting
- More -- print a string, N lines at a time
- OpenIFile, OpenOFile -- open input or output file
- OpenIString, OpenOString -- open input or output string
- OpenLog, CloseLog -- open or close a log of a CoCoA session
- Print On -- print to an output device
- Print, PrintLn -- print the value of an expression
- Source, << -- read commands from a file or device
- Sprint -- convert to a string
- Tag -- returns the tag string of an object
- Tagged, Untagged, @ -- tag or untag an object for pretty printing
- list
- .. -- range operator
- >< -- Cartesian product
- Append -- append an object to an existing list
- BlockMatrix -- create a block matrix
- BringIn -- bring in objects from another ring
- Comp -- the N-th component of a list
- Concat, ConcatLists -- concatenate lists or lists of lists, respectively
- Count -- count the objects in a list
- Diff -- returns the difference between two lists
- Distrib -- the distribution of objects in a list
- EqSet -- checks if the set of elements in two lists are equal
- First -- the first N elements of a list
- Flatten -- flatten a list
- GBM, HGBM -- intersection of ideals for zero-dimensional schemes
- GenericPoints -- random projective points
- Head -- the first element of a list
- HIntersection, HIntersectionList -- intersection of ideals
- IdealAndSeparatorsOfPoints -- ideal & separators for affine points
- IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
- IdealOfPoints -- ideal of a set of affine points
- IdealOfProjectivePoints -- ideal of a set of projective points
- In -- create a list satisfying given conditions
- Insert, Remove -- insert or remove an object in a list
- Interpolate -- interpolating polynomial
- Interreduce, Interreduced -- interreduce a list of polynomials or vectors
- Intersection, IntersectionList -- intersect lists, ideals, or modules
- IsIn -- check if one object is contained in another
- Last -- the last N elements of a list
- Len -- the length of an object
- List -- convert an expression into a list
- Mat -- convert an expression into a matrix
- Max, Min -- a maximum or minimum element of a sequence or list
- Monic -- divide polynomials by their leading coefficients
- NewList -- create a new list
- NonZero -- remove zeroes from a list
- Product, Sum -- the product or sum of the elements of a list
- Reverse, Reversed -- reverse a list
- ScalarProduct -- scalar product
- SeparatorsOfPoints -- separators for affine points
- SeparatorsOfProjectivePoints -- separators for projective points
- Set -- remove duplicates from a list
- Size -- the amount of memory used by an object
- Sort, Sorted -- sort a list
- SortBy, SortedBy -- sort a list
- Submat -- submatrix
- SubSet -- checks if the elements of one list are a subset of another
- Syz -- syzygy modules
- Tail -- remove the first element of a list
- Toric -- saturate toric ideals
- Toric.CheckInput -- check input to 'Toric'
- WithoutNth -- removes the N-th component from a list
- loops
- matrix
- Adjoint -- adjoint matrix
- BlockMatrix -- create a block matrix
- BringIn -- bring in objects from another ring
- ColumnVectors -- the list of column vectors of a matrix
- DegLexMat, DegRevLexMat, LexMat, XelMat -- matrices for std. term-orderings
- Det -- the determinant of a matrix
- Identity -- the identity matrix
- Inverse -- multiplicative inverse
- IsZero -- test whether an object is zero
- Jacobian -- the Jacobian of a list of polynomials
- Len -- the length of an object
- LinKer -- find the kernel of a matrix
- LinSol -- find a solution to a linear system
- List -- convert an expression into a list
- Mat -- convert an expression into a matrix
- Minors -- list of minor determinants of a matrix
- NewMat -- create a new matrix
- Pfaffian -- the Pfaffian of a skew-symmetric matrix
- Product, Sum -- the product or sum of the elements of a list
- Size -- the amount of memory used by an object
- Submat -- submatrix
- TensorMat -- returns the tensor product of two matrices
- Toric -- saturate toric ideals
- Toric.CheckInput -- check input to 'Toric'
- Transposed -- the transposition of a matrix
- memory
- Clear -- clear the working memory or a ring-bound memory
- Defined -- check if an expression has been defined
- Delete -- delete variables from the working memory
- Destroy -- delete rings
- GlobalMemory -- contents of global memory
- Memory -- contents of local memory or ring-bound memory
- RingEnvs -- names of all defined rings
- Size -- the amount of memory used by an object
- misc
- miscellaneous
- Block -- group several commands into a single command
- Dashes -- returns string of dashes
- Date -- the date
- Describe -- information about an expression
- Equals -- returns a string of equal signs
- Rand -- random integer
- Seed -- seed for 'Rand'
- Skip -- does nothing
- Starting -- list functions starting with a given string
- module
- Colon, :, HColon -- ideal or module quotient
- E_ -- canonical vector
- Elim -- eliminate variables
- GB.GetBettiMatrix -- returns the Betti matrix computed so far
- GB.GetNthSyz -- returns the part of the Nth syzygy module computed so far
- GB.GetNthSyzShifts -- shifts of the Nth syzygy module computed so far
- GB.GetRes -- returns the resolution computed so far
- GB.GetResLen -- returns the length of the resolution computed so far
- GB.ResReport -- status of an interactive resolution calculation
- GBasis -- calculate a Groebner basis
- GenRepr -- representation in terms of generators
- Gens -- list of generators of an ideal
- Interreduce, Interreduced -- interreduce a list of polynomials or vectors
- Intersection, IntersectionList -- intersect lists, ideals, or modules
- IsIn -- check if one object is contained in another
- IsZero -- test whether an object is zero
- Len -- the length of an object
- LT -- the leading term of an object
- Max, Min -- a maximum or minimum element of a sequence or list
- MinGens -- list minimal generators
- Minimalize, Minimalized -- remove redundant generators
- MinSyzMinGens -- minimal generators of syzygies of minimal generators
- Module -- convert an expression into a module
- NF -- normal form
- NFsAreZero -- test if normal forms are zero
- Rank -- rank of a module
- ReducedGBasis -- compute a reduced Groebner basis
- Res -- free resolution
- Size -- the amount of memory used by an object
- Syz -- syzygy modules
- SyzMinGens -- syzygy module for a set of minimal generators
- SyzOfGens -- syzygy module for a given set of generators
- online-help
- H.Browse -- browse the online help system
- H.Commands -- list commands according to type
- H.Man -- search online help system (see '?, Man')
- H.OutCommands -- print command descriptions to a file
- H.OutManual -- print the online manual to a file
- H.SetMore, H.UnSetMore -- more-device for online help
- H.Syntax -- display the syntax of a command
- H.Tips -- quick tips for using online help
- H.Toc -- display the Table Of Contents of the online manual
- H.Tutorial -- run the CoCoA tutorial
- ?, Man -- search online help system
- packages
- panels
- points
- GBM, HGBM -- intersection of ideals for zero-dimensional schemes
- GenericPoints -- random projective points
- IdealAndSeparatorsOfPoints -- ideal & separators for affine points
- IdealAndSeparatorsOfProjectivePoints -- ideal & separators for points
- IdealOfPoints -- ideal of a set of affine points
- IdealOfProjectivePoints -- ideal of a set of projective points
- Interpolate -- interpolating polynomial
- SeparatorsOfPoints -- separators for affine points
- SeparatorsOfProjectivePoints -- separators for projective points
- polynomial
- Bin -- binomial coefficient
- BringIn -- bring in objects from another ring
- Coefficients -- list of coefficients of a polynomial or vector
- CoeffOfTerm -- coefficient of a term of a polynomial or vector
- Colon, :, HColon -- ideal or module quotient
- Deg -- the degree of a polynomial or vector
- DensePoly -- the sum of all power-products of a given degree
- Der -- the derivative of a rational function
- Discriminant -- the discriminant of a polynomial
- DivAlg -- division algorithm
- Eval -- substitute numbers or polynomials for indeterminates
- Factor -- factor a polynomial
- Fraction -- returns the quotient of its arguments
- GCD, LCM -- greatest common divisor, least common multiple
- GenRepr -- representation in terms of generators
- Homogenized -- homogenize with respect to an indeterminate
- Interpolate -- interpolating polynomial
- Inverse -- multiplicative inverse
- IsHomog -- test whether given polynomials are homogeneous
- IsIn -- check if one object is contained in another
- IsTerm -- checks if the argument is a term
- IsZero -- test whether an object is zero
- Jacobian -- the Jacobian of a list of polynomials
- LC -- the leading coefficient of a polynomial or vector
- Len -- the length of an object
- LM -- the leading monomial of a polynomial or vector
- Log -- the list of exponents of the leading term of a polynomial
- LogToTerm -- returns a monomial (power-product) with given exponents
- LPP -- the leading power-product of a polynomial or vector
- LT -- the leading term of an object
- MapDown -- convert a constant polynomial to a number
- MDeg -- multi-degree of an polynomial
- Monic -- divide polynomials by their leading coefficients
- Monomials -- the list of monomials of a polynomial or vector
- NF -- normal form
- NFsAreZero -- test if normal forms are zero
- NR -- normal reduction
- Num, Den -- numerator, denominator
- Poly -- convert an expression into a polynomial
- Product, Sum -- the product or sum of the elements of a list
- QZP, ZPQ -- change field for polynomials and ideals
- Randomize, Randomized -- randomize the coefficients of a given polynomial
- Resultant -- the resultant of two polynomials
- Size -- the amount of memory used by an object
- StarPrint -- print polynomial with *'s for multiplications
- Subst -- substitute values for indeterminates
- Support -- the list of terms of a polynomial or vector
- Sylvester -- the Sylvester matrix of two polynomials
- WLog -- weighted list of exponents
- printing
- Block -- group several commands into a single command
- Close -- close a device
- Format -- convert object to formatted string
- Get -- read characters from a device
- IO.SprintTrunc -- convert to a string and truncate
- Latex -- LaTeX formatting
- More -- print a string, N lines at a time
- OpenIFile, OpenOFile -- open input or output file
- OpenIString, OpenOString -- open input or output string
- OpenLog, CloseLog -- open or close a log of a CoCoA session
- Print On -- print to an output device
- Print, PrintLn -- print the value of an expression
- Source, << -- read commands from a file or device
- Sprint -- convert to a string
- Tag -- returns the tag string of an object
- Tagged, Untagged, @ -- tag or untag an object for pretty printing
- programming
- quotient
- Dim -- the dimension of a ring or quotient object
- GBasis -- calculate a Groebner basis
- Hilbert -- the Hilbert function
- HilbertPoly -- the Hilbert polynomial
- HVector -- the h-vector of a ring or quotient object
- MinGens -- list minimal generators
- MinSyzMinGens -- minimal generators of syzygies of minimal generators
- Multiplicity -- the multiplicity (degree) of a ring or quotient object
- Poincare, HilbertSeries -- the Poincare series
- ReducedGBasis -- compute a reduced Groebner basis
- Res -- free resolution
- SyzMinGens -- syzygy module for a set of minimal generators
- SyzOfGens -- syzygy module for a given set of generators
- rat
- Abs -- absolute value of a number
- FloatStr, MantissaAndExponent -- convert rational number to a float string
- Fraction -- returns the quotient of its arguments
- Inverse -- multiplicative inverse
- IsNumber -- checks if the argument is a number
- IsZero -- test whether an object is zero
- Max, Min -- a maximum or minimum element of a sequence or list
- Num, Den -- numerator, denominator
- Product, Sum -- the product or sum of the elements of a list
- ratfun
- BringIn -- bring in objects from another ring
- Der -- the derivative of a rational function
- Fraction -- returns the quotient of its arguments
- Inverse -- multiplicative inverse
- IsZero -- test whether an object is zero
- Len -- the length of an object
- Num, Den -- numerator, denominator
- Product, Sum -- the product or sum of the elements of a list
- Size -- the amount of memory used by an object
- Subst -- substitute values for indeterminates
- rational
- record
- ring
- Characteristic -- the characteristic of a ring
- Clear -- clear the working memory or a ring-bound memory
- CurrentRing -- the current ring
- Deg -- the degree of a polynomial or vector
- Delete -- delete variables from the working memory
- Destroy -- delete rings
- Dim -- the dimension of a ring or quotient object
- Hilbert -- the Hilbert function
- HilbertPoly -- the Hilbert polynomial
- HVector -- the h-vector of a ring or quotient object
- Image -- ring homomorphism
- Indet -- individual indeterminates
- IndetInd -- the index of an indeterminate
- IndetIndex -- index of an indeterminate
- Indets -- list of current indeterminates
- IndetName -- the name of an indeterminate
- IsPosTo, IsToPos -- checks the module term-ordering of a ring
- MDeg -- multi-degree of an polynomial
- Multiplicity -- the multiplicity (degree) of a ring or quotient object
- NumIndets -- number of indeterminates
- Ord -- matrix defining a term-ordering
- Poincare, HilbertSeries -- the Poincare series
- QZP, ZPQ -- change field for polynomials and ideals
- Ring -- returns the ring with a given name
- RingEnv -- name of the current ring
- RingEnvs -- names of all defined rings
- TypeOfCoeffs -- type of the coefficients of the current ring
- Use -- command for making a ring active
- Using -- perform commands in non-active ring
- WeightsList -- first row of the weights matrix
- WeightsMatrix -- matrix of generalized weights for indeterminates
- string
- Ascii -- convert between characters and ascii code
- Comp -- the N-th component of a list
- FloatStr, MantissaAndExponent -- convert rational number to a float string
- Format -- convert object to formatted string
- IO.SprintTrunc -- convert to a string and truncate
- IsIn -- check if one object is contained in another
- Latex -- LaTeX formatting
- Max, Min -- a maximum or minimum element of a sequence or list
- More -- print a string, N lines at a time
- NewId -- create a new identifier
- OpenIString, OpenOString -- open input or output string
- Spaces -- return a string of spaces
- Sprint -- convert to a string
- Var -- function calls by reference, other complex referencing
- system
- Ciao -- quit CoCoA
- CocoaLibrary -- returns the path to the CoCoA libraries
- CocoaLimits -- limits on exponents and ring characteristics
- Contrib -- list of contributors to CoCoA
- GlobalMemory -- contents of global memory
- More -- print a string, N lines at a time
- Option -- status of a panel option
- Panel -- print status of a panel's options
- Panels -- list of CoCoA panels
- Quit -- quit CoCoA
- Reset -- reset panels and random number seed to defaults
- ResetPanels -- reset panels to their default values
- Set, Unset -- set and unset panel options
- tags
- toric
- type
- vector
- BringIn -- bring in objects from another ring
- CoeffOfTerm -- coefficient of a term of a polynomial or vector
- Colon, :, HColon -- ideal or module quotient
- ColumnVectors -- the list of column vectors of a matrix
- Comp -- the N-th component of a list
- Comps -- list of components of a vector
- Deg -- the degree of a polynomial or vector
- DivAlg -- division algorithm
- E_ -- canonical vector
- FirstNonZero, FirstNonZeroPos -- the first non-zero entry in a vector
- GenRepr -- representation in terms of generators
- IsIn -- check if one object is contained in another
- IsTerm -- checks if the argument is a term
- IsZero -- test whether an object is zero
- LC -- the leading coefficient of a polynomial or vector
- Len -- the length of an object
- List -- convert an expression into a list
- LM -- the leading monomial of a polynomial or vector
- LPos -- the position of the leading power-product in a vector
- LPP -- the leading power-product of a polynomial or vector
- LT -- the leading term of an object
- Mat -- convert an expression into a matrix
- Monomials -- the list of monomials of a polynomial or vector
- NewVector -- create a new vector
- NF -- normal form
- NFsAreZero -- test if normal forms are zero
- NonZero -- remove zeroes from a list
- NR -- normal reduction
- NumComps -- the number of components of a vector
- Product, Sum -- the product or sum of the elements of a list
- ScalarProduct -- scalar product
- Size -- the amount of memory used by an object
- Support -- the list of terms of a polynomial or vector
- Vector -- create a vector
- zmod
CoCoA Commands
..
SYNTAX
M .. N
where M and N are of type INT or M and N are indeterminates of the
current ring.
SUMMARY
range operator
DESCRIPTION
If M and N are of type INT, then the expression:
M .. N
returns
* the list [M, M+1, ... ,N] if M <= N;
* the empty list, [], otherwise.
Note: Large values for M and N are not permitted; typically
they should lie in the range about -10^9 to +10^9.
If x and y are indeterminates in a ring, then
x .. y
gives the indeterminates lying between x and y in the order
they appear in the definition of the ring.
>EXAMPLE<
1..10;
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
-------------------------------
Use R ::= Q[x,y,z,a,b,c,d];
z..c;
[z, a, b, c]
-------------------------------
See also:
CoCoA Operators
><
SYNTAX L_1 >< ... >< L_n where each L_i is a list. SUMMARY Cartesian product DESCRIPTION This operator (made using a greater-than sign '>' and a less-than sign '<') returns the list whose elements form the Cartesian product of L_1,...,L_n. >EXAMPLE< L := [1,2,3]; L >< ["a","b"] >< [5]; [[1, "a", 5], [1, "b", 5], [2, "a", 5], [2, "b", 5], [3, "a", 5], [3, "b", 5]] ------------------------------- Note that only '<>' is used for 'not equal' in CoCoA. See also: CoCoA Operators
Abs
SYNTAX Abs(N:INT):INT Abs(N:RAT):RAT SUMMARY absolute value of a number DESCRIPTION This function returns the absolute value of N. >EXAMPLE< Abs(-3); 3 ------------------------------- Abs(-2/3); 2/3 -------------------------------
Adjoint
SYNTAX Adjoint(M:MAT):MAT where M is a square matrix. SUMMARY adjoint matrix DESCRIPTION This function returns the adjoint matrix of M. >EXAMPLE< Adjoint(Mat([[x,y,z],[t,y,x],[x,x^2,xy]])); Mat[ [-x^3 + xy^2, -xy^2 + x^2z, xy - yz], [-txy + x^2, x^2y - xz, -x^2 + tz], [tx^2 - xy, -x^3 + xy, -ty + xy] ] ------------------------------- Adjoint(Mat([[1%5,2%5],[3%5,1%5]])); Mat[ [1 % 5, -2 % 5], [2 % 5, 1 % 5] ] -------------------------------
Alias
SYNTAX Alias B_1,..., B_r where each B_i is a 'binding' of the form: Identifier := $PackageName SUMMARY define aliases for package names DESCRIPTION This function is for declaring both global and local aliases for package names. Recall that package names are meant to be long in order to avoid conflicts between the names of functions that are read into a CoCoA session. However, it is inconvenient to have to type out the long package name when referencing a function. So the user chooses an alias to take the place of the package name; the alias is just a means to avoid typing. Aliases for packages that are routinely loaded, may be added to 'userinit.coc' (see 'User Initialization'). 1. Global aliases. To avoid typing the full package name as a prefix to package functions, one may declare a short global alias during a CoCoA session. A list of the global aliases is produced by the function 'Aliases'. For examples, see the chapter on packages in the manual, in particular the section, 'Global Aliases'. Online, enter '?global aliases'. 2. Local aliases. A local alias has the same syntax as a global alias, however it appears inside a package definition. The local aliases work only inside the package and do not conflict with any global aliases already defined. In fact, in order to avoid conflicts, global aliases are not recognized within a package. For examples, again look in the chapter for packages. >EXAMPLE< Alias LL := $abcd; Aliases(); H = $cocoa/help IO = $cocoa/io GB = $cocoa/gb HP = $cocoa/hp HL = $cocoa/hilop List = $cocoa/list Mat = $cocoa/mat Latex = $cocoa/latex LaTeX = $cocoa/latex Toric = $cocoa/toric Coclib = $cocoa/coclib ------------------------------- See also: Alias In Aliases Global Aliases Introduction to Packages Local Aliases
Alias In
SYNTAX
Alias B_1,...,B_r In C EndAlias
where each B_i is a 'binding' of the form: Identifier := $PackageName,
and C is a command sequence.
SUMMARY
temporarily override global aliases
DESCRIPTION
This command allows one to use the aliases defined by the B_i's in
the command sequence C without affecting the global aliases.
>EXAMPLE<
Aliases(); -- the global aliases
H = $cocoa/help
IO = $cocoa/io
GB = $cocoa/gb
HP = $cocoa/hp
HL = $cocoa/hilop
List = $cocoa/list
Mat = $cocoa/mat
Latex = $cocoa/latex
LaTeX = $cocoa/latex
Toric = $cocoa/toric
Coclib = $cocoa/coclib
-------------------------------
Alias HP := $cocoa/help In HP.Man('Alias In') EndAlias;
============ Alias In =============
SYNTAX
Alias B_1,...,B_r In C EndAlias
where each B_i is a 'binding' of the form: Identifier := $PackageName,
and C is a command sequence.
---> Output suppressed <---
HP.Examples(); -- the global alias HP is unaffected
---- STANDARD ----
Use R ::= Q[txyz];
Poincare(R);
---> Output suppressed <---
See also:
Alias
Introduction to Packages
Aliases
SYNTAX
Aliases():TAGGED("Aliases")
SUMMARY
list of global aliases
DESCRIPTION
This function prints a list of global aliases for packages. Aliases
are formed with the function 'Alias'.
>EXAMPLE<
Aliases();
H = $cocoa/help
IO = $cocoa/io
GB = $cocoa/gb
HP = $cocoa/hp
HL = $cocoa/hilop
List = $cocoa/list
Mat = $cocoa/mat
Latex = $cocoa/latex
LaTeX = $cocoa/latex
Toric = $cocoa/toric
Coclib = $cocoa/coclib
-------------------------------
Alias TT := $abc;
Aliases();
H = $cocoa/help
IO = $cocoa/io
GB = $cocoa/gb
HP = $cocoa/hp
HL = $cocoa/hilop
List = $cocoa/list
Mat = $cocoa/mat
Latex = $cocoa/latex
LaTeX = $cocoa/latex
Toric = $cocoa/toric
Coclib = $cocoa/coclib
TT = $abc
-------------------------------
See also:
Alias
Introduction to Packages
Ascii
SYNTAX
Ascii(N:INT):STRING
Ascii(L:LIST of INT):STRING
Ascii(S:STRING):LIST of INT
SUMMARY
convert between characters and ascii code
DESCRIPTION
In the first form, Ascii returns the character whose ascii code is N.
In the second form, Ascii returns the string whose characters, in
order, have the ascii codes listed in L.
The third form is the inverse of the second: it returns the ascii codes
of the characters in S.
>EXAMPLE<
Ascii(97);
a
-------------------------------
C := Ascii('hello world');
C;
[104, 101, 108, 108, 111, 32, 119, 111, 114, 108, 100]
-------------------------------
Ascii(C);
hello world
-------------------------------
Append
SYNTAX Append(V:LIST, E:OBJECT):NULL where V is a variable containing a list. SUMMARY append an object to an existing list DESCRIPTION This function appends the object E to the list V. >EXAMPLE< Use R ::= Q[t,x,y,z]; L := [1,2,3]; Append(L,4); L; [1, 2, 3, 4] ------------------------------- See also: Concat Remove
Bin
SYNTAX
Bin(N:INT or POLY,K:INT):INT
SUMMARY
binomial coefficient
DESCRIPTION
This function computes the binomial coefficient, 'N choose K'
according to the formula
(N)(N-1)(N-2)...(N-K+1)/ K!
The same formula is used if N is a polynomial. The integer K cannot
be negative.
>EXAMPLE<
Bin(4,2);
6
-------------------------------
Bin(-4,3);
-20
-------------------------------
Bin(x^2+2y,3);
1/6x^6 + x^4y - 1/2x^4 + 2x^2y^2 - 2x^2y + 4/3y^3 + 1/3x^2 - 2y^2 + 2/3y
-------------------------------
It = (x^2+2y)(x^2+2y-1)(x^2+2y-2)/6;
TRUE
-------------------------------
See also:
BinExp
BinExp, EvalBinExp
SYNTAX
BinExp(N:INT,K:INT):TAGGED("$cocoa/binrepr.BinExp")
BinExp(N:INT,K:INT,Up:INT,Down:INT):INT
EvalBinExp(B:TAGGED("$cocoa/binrepr.BinExp"),Up:INT,Down:INT):INT
where N and K are positive integers, and Up and Down are integers.
SUMMARY
binomial expansion
DESCRIPTION
The first function computes the K-binomial expansion of N, i.e., the
unique expression
N = Bin(N(K),K) + Bin(N(K-1),K-1) + ... + Bin(N(I),I)
where N(K) > ... > N(I) >= 1, for some I. The value returned is
tagged for pretty printing.
The second function computes the sum of the binomial coefficients
appearing in the K-binomial expansion of N after replacing each
summand Bin(N(J),J) by Bin(N(J)+Up,J+Down). It is useful in
generalizations of Macaulay's theorem characterizing Hilbert
functions.
The third function computes the same integer as the second except it
accepts BinExp(N,K) as an argument rather than N and K.
>EXAMPLE<
BE := BinExp(13,4);
BE;
Bin(5,4) + Bin(4,3) + Bin(3,2) + Bin(1,1)
-------------------------------
EvalBinExp(BE,1,1);
16
-------------------------------
BinExp(13,4,1,1);
16
-------------------------------
EvalBinExp(BE,0,0); -- the integer value of BE
13
-------------------------------
See also:
Bin
Block
SYNTAX Block C_1; ... ; C_n EndBlock; where each C_i is a command. SUMMARY group several commands into a single command DESCRIPTION The 'Block' command executes the commands as if they where one command. What this means in practice is that CoCoA will not print a string of dashes after executing each C_i. Thus, 'Block' is used on-the-fly and not inside user-defined functions. (It has nothing to do with declaration of local variables, for instance, as one might infer from some other computer languages.) The following example should make the use of 'Block' clear: >EXAMPLE< Print 'hello '; Print 'world'; hello ------------------------------- world ------------------------------- Block Print 'hello '; Print 'world' EndBlock; hello world ------------------------------- Block PrintLn GCD([12,24,96]); PrintLn LCM([12,24,96]); PrintLn GCD([x+y,x^2-y^2]); Print LCM([x+y,x^2-y^2]); EndBlock; 12 96 x + y x^2 - y^2 -------------------------------
BlockMatrix
SYNTAX BlockMatrix(L:LIST):MAT where L is a list representing a block matrix. SUMMARY create a block matrix DESCRIPTION This function creates a block matrix. Each entry of the input list L has the form [M_1,...,M_k] where each M_i is either: (i) a matrix (or list cast-able to a matrix) or (ii) the number 0, representing a zero matrix of arbitrary size. The entry represents a row of a block matrix. For instance, if A, B, C, and D are matrices, then BlockMatrix([A,B,0],[C,0,D]] will return a matrix of the form | A B 0 | | C 0 D |. The obvious restrictions on the sizes of the matrices apply. In the above example, we would need the number of rows in A and B to be the same. Similarly for C and D. The number of columns in A and C would need to be the same. >EXAMPLE< A := [[1,2,3],[4,5,6]]; B := [[1,2],[3,4]]; C := [[1,1,1],[2,2,2],[3,3,3]]; D := [[4,4],[5,5],[6,6]]; BlockMatrix([[A,B,0],[C,0,D]]); Mat[ [1, 2, 3, 1, 2, 0, 0], [4, 5, 6, 3, 4, 0, 0], [1, 1, 1, 0, 0, 4, 4], [2, 2, 2, 0, 0, 5, 5], [3, 3, 3, 0, 0, 6, 6] ] -------------------------------
Break
SYNTAX
Break
SUMMARY
break out of a loop
DESCRIPTION
This command must be used inside a loop statement ('For', 'Foreach',
'Repeat', or 'While'). When executed, the current loop statement is
terminated and control passes to the command following the loop
statement. Thus, in the case of nested loops 'Break' does *not* break
out of all loops back to the "top level" (see 'Return').
>EXAMPLE<
For I := 5 To 1 Step -1 Do
For J := 1 To 100 Do
Print J,' ';
If J = I Then PrintLn; Break EndIf;
EndFor;
EndFor;
1 2 3 4 5
1 2 3 4
1 2 3
1 2
1
-------------------------------
See also:
Return
BringIn
SYNTAX BringIn(E:OBJECT):OBJECT where E is a polynomial, a rational function, or a list/matrix/vector of these. SUMMARY bring in objects from another ring DESCRIPTION This function maps a polynomial or rational function (or a list, matrix, or vector of these) into the current ring, preserving the names of the indeterminates. When mapping from a ring of finite characteristic to one of zero characteristic then consistent choices of image for the coefficients are made (i.e. if two coefficients are equal mod p then their images will be equal). If the two polynomial rings differ only in characteristic then it is faster to use the function QZP or ZPQ. >EXAMPLE< RR ::= Q[x[1..4],z,y]; SS ::= Z/(101)[z,y,x[1..2]]; Use RR; F := (x[1]-y-z)^2; F; x[1]^2 - 2x[1]z + z^2 - 2x[1]y + 2zy + y^2 ------------------------------- Use SS; B := BringIn(F); B; z^2 + 2zy + y^2 - 2zx[1] - 2yx[1] + x[1]^2 ------------------------------- Use R::=Q[x,y,z]; F := 1/2*x^3 + 34/567*x*y*z - 890; -- a poly with rational coefficients Use S::=Z/(101)[x,y,z]; QZP(F) = BringIn(F); TRUE -------------------------------
Call
SYNTAX
Call(F:FUNCTION,X_1,...,X_n):OBJECT
where X_1,...,X_n are the arguments for the function F.
SUMMARY
apply a function to given arguments
DESCRIPTION
This function applies the function F to the arguments X_1,...X_n.
>EXAMPLE<
The following function MyMax takes a function LessThan as parameter,
and returns the maximum of X and Y w.r.t. the ordering defined by the
function LessThan.
Define MyMax(LessThan,X,Y)
If Call(LessThan,X,Y) Then Return Y Else Return X EndIf
EndDefine;
Let's use MyMax by giving two different orderings.
Define LT_Standard(X,Y)
Return X < Y;
EndDefine;
Define LT_First(X,Y)
Return TRUE;
End;
MyMax(Function("LT_Standard"),3,5);
5
-------------------------------
MyMax(Function("LT_First"),5,3);
3
-------------------------------
MyMax(Function("LT_First"),3,5);
5
-------------------------------
MyMax(Function("LT_First"),5,3);
3
-------------------------------
See also:
Function
Cast
SYNTAX
Cast(E:OBJECT,T:TYPE):TYPE
SUMMARY
type conversion
DESCRIPTION
This function returns the value of the expression E after converting
it to type T. If S and T are types with S < T, then casting from S to
T is usually possible.
>EXAMPLE<
L := [[1,2],[3,4]];
Type(L);
LIST
-------------------------------
Cast(L,MAT);
Mat[
[1, 2],
[3, 4]
]
-------------------------------
L; -- L is unchanged; it is still a list.
[[1, 2], [3, 4]]
-------------------------------
Use Z/(5)[t];
A := 8;
A; -- A has type INT
8
-------------------------------
Cast(A,POLY); -- cast as a polynomial, A = -2 since the coefficient
-- ring is Z/5Z
-2
-------------------------------
See also:
Data Types
Shape
Type
Types
Catch
SYNTAX
Catch C EndCatch;
Catch C In E EndCatch;
where C is a sequence of commands and E is a variable identifier.
SUMMARY
catch an error
DESCRIPTION
Usually, when an error occurs during the execution of a command, the
error is automatically propagated out of the nesting of the
evaluation. This can be prevented with the use of 'Catch'.
If an error occurs during the execution of C, then it is captured by
the command 'Catch' and (in the second form) assigned to the variable
E. If no error occurs, then E will contain the value 'Null'. Note
the use of the function 'GetErrMesg' in the example below.
IMPORTANT NOTE: There is a bug in 'Catch'. Any 'Return' command used
inside 'Catch' must return some value. If not, the 'Return' command
will just return from the Catch-EndCatch statement; it will not return from
the function within which the statement is embedded. There is an
example below.
>EXAMPLE<
Define Test(N)
Catch
PrintLn(1/N);
In E EndCatch;
If Type(E) = ERROR Then Print("An error occurred: ", GetErrMesg(E)) EndIf;
EndDefine;
Test(3);
1/3
-------------------------------
Test(0);
An error occurred: Division by zero
-------------------------------
--Illustration of the BUG --
Define Test2()
Catch
Print('Hello ');
Return; -- incorrect: no value is returned
EndCatch;
PrintLn('world.');
EndDefine;
Test2();
Hello world.
-------------------------------
Define Test3()
Catch
Print('Hello ');
Return 3; -- correct a value is returned
EndCatch;
PrintLn('world.');
EndDefine;
Test3();
Hello 3
-------------------------------
See also:
Error
GetErrMesg
Characteristic
SYNTAX Characteristic():INT Characteristic(R:RING):INT SUMMARY the characteristic of a ring DESCRIPTION This function returns the characteristic of the current ring, in the first case, or of the ring R, in the second. >EXAMPLE< Use R ::= Z/(3)[t]; S ::= Q[x,y]; Characteristic(); -- characteristic of the current ring, R 3 ------------------------------- Characteristic(S); 0 -------------------------------
Ciao
SYNTAX Ciao SUMMARY quit CoCoA DESCRIPTION This command is used to quit CoCoA. Note, it is issued as follows: Ciao; without parentheses. See also: Quit
Clear
SYNTAX
Clear
Clear R_1,...,R_n
where the R_i are identifiers for rings.
SUMMARY
clear the working memory or a ring-bound memory
DESCRIPTION
The first form clears the working memory, i.e, all non-global
variables. In the second form, the command clears the global
variables bound to the rings R_1,...,R_n, i.e., the "ring-bound"
memory for these rings. For more information on memory in CoCoA, see
the chapter entitled "Memory Management."
The contents of the working memory are listed by the command
'Memory()', and the global variables bound to the ring R are listed by
the command 'Memory(R)'.
>EXAMPLE<
Use R ::= Q[x,y,z];
I := Ideal(x,y); -- I is added to the working memory
MEMORY.X := 3; -- a global variable
ENV.R.X := Ideal(x); -- a global variable bound to the ring R
-- note that 'ENV' is equivalent to 'MEMORY.ENV'
Use S ::= Q[a,b];
ENV.S.Y := Ideal(a^2); -- global variable bound to S
J := Ideal(a,b); -- J is added to the working memory
Z := 4; -- Z is added to the working memory
Memory(); -- the contents of the working memory
["I", "J", "UserInitFile", "Z"]
-------------------------------
Memory(R); -- the global variables bound to R
["X"]
-------------------------------
Memory(S); -- the global variables bound to S
["Y"]
-------------------------------
Clear; -- clear the working memory
Memory();
[ ]
-------------------------------
Clear R; -- clear the global variables bound to R
Memory(R);
[ ]
-------------------------------
Memory(S);
["Y"]
-------------------------------
ENV.S.Y; -- this variable was never cleared
Ideal(a^2)
-------------------------------
See also:
Delete
Destroy
Memory
Memory Management
Close
SYNTAX
Close(D:DEVICE)
SUMMARY
close a device
DESCRIPTION
This function closes the device D.
>EXAMPLE<
D := OpenOFile('my-test'); -- open file for output from CoCoA
Print 'test' On D; -- write to my-file
Close(D); -- close the file
Close(DEV.STDIN); -- close the standard input device
-- Bye
(Close(DEV.OUT) suppresses all output to the CoCoA window.)
See also:
Introduction to IO
CocoaLibrary
SYNTAX CocoaLibrary():STRING SUMMARY returns the path to the CoCoA libraries DESCRIPTION This function returns the path name of the directory containing the CoCoA libraries. It is platform dependent. >EXAMPLE< CocoaLibrary(); /usr/local/cocoa-4.1 -------------------------------
CocoaLimits
SYNTAX CocoaLimits():RECORD SUMMARY limits on exponents and ring characteristics DESCRIPTION This function returns the maximum allowable characteristic of a CoCoA ring and the maximum allowable exponent in a CoCoA expression. These numbers may vary depending on the platform on which CoCoA is run. >EXAMPLE< CocoaLimits(); Record[MaxChar = 32767, MaxExp = 2147483647] -------------------------------
Coefficients
SYNTAX Coefficients(F:POLY or VECTOR):LIST Coefficients(F:POLY,X:INDET):LIST Coefficients(F:POLY,S:LIST):LIST SUMMARY list of coefficients of a polynomial or vector DESCRIPTION This function returns the coefficients of F. In the first form, a list of the (non-zero) coefficients is returned; the order being decreasing on the terms in F as determined by the term-ordering of the ring to which F belongs. In the second form, the function views F as a polynomial in X, and returns a list of coefficients which are polynomials in the remaining variables; their order is decreasing in powers of X, and a zero value is given for those powers of X absent from F. In the third form, the coefficients of the specified terms are returned; their order is determined by the list S. >EXAMPLE< Use R ::= Q[x,y,z]; F := 3x^2y+5y^3-xy^5; Coefficients(F); [-1, 3, 5] ------------------------------- ScalarProduct(Coefficients(F),Support(F)) = F; TRUE ------------------------------- V:=Vector(3x^2+y,x-5z^3); Coefficients(V); [-5, 3, 1, 1] ------------------------------- ScalarProduct(Coefficients(V),Support(V))=V; TRUE ------------------------------- Coefficients(x^3z+xy+xz+y+2z,x); [z, 0, y + z, y + 2z] ------------------------------- F := (1+2*x+3*y^4+5*z^6)^7; Skeleton := [1,x^3,y^12,z^19,x^2*y^8*z^12]; Coefficients(F, Skeleton); [1, 280, 945, 0, 567000] ------------------------------- See also: Coefficient Rings LC Monomials Support
CoeffOfTerm
SYNTAX CoeffOfTerm(T:POLY,F:POLY):C CoeffOfTerm(T:VECTOR,F:VECTOR):C where T is a term (no coefficient) and C is one of INT, RAT, or ZMOD. SUMMARY coefficient of a term of a polynomial or vector DESCRIPTION This function returns the coefficient of the term T occurring in F. >EXAMPLE< Use R ::= Q[x,y,z]; F := 5xy^2-3z^3; CoeffOfTerm(xy^2,F); 5 ------------------------------- CoeffOfTerm(x^3,F); 0 ------------------------------- CoeffOfTerm(z^3,F); -3 ------------------------------- CoeffOfTerm(Vector(0,x^3,0),Vector(x+3,6xy-5x^3,x-z^2)); -5 ------------------------------- See also: Coefficients LC Log LogToTerm Monomials Support
Colon, :, HColon
SYNTAX Colon(M:IDEAL,N:IDEAL):IDEAL Colon(M:MODULE,N:MODULE):IDEAL M : N HColon(M:IDEAL,N:IDEAL):IDEAL SUMMARY ideal or module quotient DESCRIPTION These functions return the quotient of M by N: the ideal of polynomials F in R such that FG is in M for all G in N. The command 'M : N' is a shortcut for 'Colon(M,N)'. The function 'HColon' calculates the ideal quotient using a Hilbert-driven algorithm. It differs from 'Colon' only when the input is inhomogeneous, in which case, 'HColon' may be faster. >EXAMPLE< Use R ::= Q[x,y]; Ideal(xy,x^2) : Ideal(x); Ideal(y, x) ------------------------------- Colon(Ideal(x^2,xy),Ideal(x,x-y^2)); Ideal(x) ------------------------------- HColon(Ideal(x^2,xy),Ideal(x,x-y^2)); Ideal(x) ------------------------------- See also: HSaturation Saturation
ColumnVectors
SYNTAX ColumnVectors(M:LIST or MAT):LIST of VECTOR where if M is a list, is must be cast-able as a matrix. SUMMARY the list of column vectors of a matrix DESCRIPTION This function returns the list of column vectors of the matrix M. >EXAMPLE< Use R ::= Q[x,y]; M := Mat([[1,1],[x,y],[x^2,y^2]]); M; Mat[ [1, 1], [x, y], [x^2, y^2] ] ------------------------------- ColumnVectors(M); [Vector(1, x, x^2), Vector(1, y, y^2)] -------------------------------
Comp
SYNTAX
Comp(E:LIST, RECORD, STRING, or VECTOR,X_1:INT,...,X_k:INT):OBJECT
SUMMARY
the N-th component of a list
DESCRIPTION
This function returns E[X_1,...,X_k] except in the case where there
are no additional arguments X_1,...,X_k, in which case E, itself, is
returned (in other words 'Comp(E)' returns E).
>EXAMPLE<
Use R ::= Q[x,y,z];
L := [4,5,[6,7],8];
Comp(L,1);
4
-------------------------------
Comp(L,3);
[6, 7]
-------------------------------
Comp(L,3,2);
7
-------------------------------
F(X):=[X,X^2]; -- the following usage of 'Comp' is useful for
-- programming
F(2);
[2, 4]
-------------------------------
Comp(F(2),2);
4
-------------------------------
Struct := Record[L := [x,y,z], S := "string"];
Struct["L",3]; -- 'Comp' works for records also
z
-------------------------------
Comp(Struct,"L",3);
z
-------------------------------
Comp("this is a string",3); -- use of 'Comp' with strings
i
-------------------------------
Comps
SYNTAX Comps(V:VECTOR):LIST SUMMARY list of components of a vector DESCRIPTION This function returns the list of components of V. It is the same as Cast(V,LIST). >EXAMPLE< Use R ::= Q[x,y]; Comps(Vector(x,x+y,x+y^2)); [x, x + y, y^2 + x] ------------------------------- See also: Comp NumComps
Concat, ConcatLists
SYNTAX Concat(L_1:LIST,...,L_n:LIST):LIST ConcatLists([L_1:LIST,...,L_n:LIST]):LIST SUMMARY concatenate lists or lists of lists, respectively DESCRIPTION The first function returns the list obtained by concatenating lists L_1,...,L_n. >EXAMPLE< Concat([1,2,3],[4,5],[],[6]); [1, 2, 3, 4, 5, 6] The second function takes a list whose components are lists and returns the concatenation of these components. >EXAMPLE< L := [[1,2],["abc","def"],[3,4]]; ConcatLists(L); [1, 2, "abc", "def", 3, 4] ------------------------------- See also: Concatenation
Cond
SYNTAX Cond B_1 Then E_1 EndCond Cond B_1 Then E_1 Elsif B_2 Then E_2 Elsif ... EndCond Cond B_1 Then E_1 Elsif B_2 Then E_2 Elsif ... Else E_r EndCond Cond(B_1,E_1,B_2,E_2,...,E_r) where the B_i's are boolean expressions and the E_i's are expressions. SUMMARY conditional expression DESCRIPTION If B_n is the first in the sequence of B_i's to evaluate to TRUE, then E_n is returned. If none of the B_i's evaluates to TRUE, then Null is returned. The construct, 'Elsif B Then E' can be repeated any number of times. Note: be careful not to type 'Elseif' by mistake (it has an extraneous 'e'). The difference between 'Cond' and 'If' is that 'Cond' is an expression which may be assigned to a variable; each of the E_i's is an expression, not a general sequence of commands (as their analogues in 'If' might be). >EXAMPLE Define Sign(A) Return Cond A>0 Then 1 Elsif A=0 Then 0 Else -1 EndCond; EndDefine; Sign(3); 1 ------------------------------- Define PrintSign(A) Return Cond(A>0,"positive",A=0,"zero","negative"); End; PrintSign(3); positive ------------------------------- See also: If
Contrib
SYNTAX Contrib():NULL SUMMARY list of contributors to CoCoA DESCRIPTION This function returns a list of contributors to the main CoCoA system. In addition to these contributions, there are many other contributions which are not part of the standard distribution. For pointers to these, see the CoCoA homepage at http://cocoa.dima.unige.it or one of its mirrors.
Count
SYNTAX Count(L:LIST,E:OBJECT):INT SUMMARY count the objects in a list DESCRIPTION This function counts the number of occurrences of the object E in the list L. >EXAMPLE< L := [1,2,3,2,[2,3]]; Count(L,2); 2 ------------------------------- Count(L,[2,3]); 1 ------------------------------- Count(L,"a"); 0 ------------------------------- See also: Distrib Len
CurrentRing
SYNTAX CurrentRing() SUMMARY the current ring DESCRIPTION This function returns the current ring. The related command, 'RingEnv' returns the name of the current ring. >EXAMPLE< Use R ::= Q[x,y]; Use S ::= Z/(3)[t]; CurrentRing(); Z/(3)[t] ------------------------------- Use R; CurrentRing(); Q[x,y] ------------------------------- See also: Ring RingEnv RingEnvs
Dashes
SYNTAX Dashes() SUMMARY returns string of dashes DESCRIPTION This function returns a string of dashes: >EXAMPLE< Dashes(); 1+1; Dashes(); 1+1; ------------------------------- ------------------------------- 2 ------------------------------- See also: Equals
Date
SYNTAX Date() SUMMARY the date DESCRIPTION This function returns the date. >EXAMPLE< Date(); Fri Jan 30 20:47:18 1998 -------------------------------
Define
SYNTAX
Define F(X_1,...,X_n) Help S:STRING; C EndDefine
F(X_1,...,X_n) := E
Define F(...) Help S:STRING; C EndDefine
where F is an identifier, C is a sequence of commands, the X_i's are
formal parameters and E is an expression. The third form, which
literally includes the string '...' is used for a variable number of
parameters. The optional 'Help S', where S is a string, may be added
to provide help for the user.
SUMMARY
define a function
DESCRIPTION
1. INTRODUCTION. This command adds the user-defined function F to the
library. The function F can be called in the following way:
F(E_1,...,E_n)
where the E_i's are expressions. The result of the evaluation of each
expression E_i is assigned to the respective formal parameter X_i, and
the command sequence C is executed. If, during the execution of C, a
statement 'Return E' is executed, then the result of the evaluation of
E is the return-value of the function F. If no 'Return' command is
executed, or 'Return' is executed without argument, then the
return-value is 'Null'.
>EXAMPLE<
Define Square(X)
Return X^2;
EndDefine;
Square(5);
25
-------------------------------
2. SCOPE. Every variable defined or modified by the command sequence
C is considered local to the function unless the variable is global or
relative to a 'Var' parameter. For the use of global variables, see
'Global Memory' or the example below. See 'Var' to learn about
calling a function 'by reference', i.e. so that the function can
change the value of an existing variable.
>EXAMPLE<
Define Example_1(L)
L := L + 5;
Return L;
EndDefine;
L := 0;
Example_1(L);
5
-------------------------------
L; -- L is unchanged despite the function call.
0
-------------------------------
Define Example_2(L) -- Example using a global variable.
MEMORY.X := L + 3;
EndDefine;
Example_2(10);
MEMORY.X;
13
-------------------------------
3. VARIABLE NUMBER OF PARAMETERS. It is also possible to have a
variable number of parameters using the syntax
Define F(...) Help S:STRING; C EndDefine
In this case the special variable ARGV will contain the list of
the arguments passed to the function. (The statement, 'Help S;' is
optional.)
Define Sum(...)
If Len(ARGV) = 0 Then Return Null; -- empty sum
Else
Sum := 0;
Foreach N In ARGV Do Sum := Sum+N EndForeach;
EndIf;
Return Sum;
EndDefine;
Sum(1,2,3,4,5);
15
-------------------------------
Sum();
Null
-------------------------------
NOTE: The use of the following shorthand is discouraged and may be
discontinued in later versions of CoCoA.
4. SHORTCUT. The form 'F(X_1,...,X_n) := E' is shorthand for 'Define
F(X_1,...X_n) Return E EndDefine;
>EXAMPLE<
F(X) := X^2;
F(5);
25
-------------------------------
5. HELP. Inside a user-defined function, one may add the command:
Help S;
where S is a string. Then, when a user enters 'Help('F')' where F is
the identifier for the function, the string, S, is printed.
>EXAMPLE<
Define Test(N)
Help 'Usage: Test(N:INT):INT';
Return N;
EndDefine;
Help 'Test';
Usage: Test(N:INT):INT
-------------------------------
6. DEFINING RINGS INSIDE FUNCTIONS. For information on this topic,
please see the section of the tutorial entitled, "Rings Inside
User-Defined Functions"
See also:
An Overview of CoCoA Programming
Introduction to User-Defined Functions
Memory Management
Return
Rings Inside User-Defined Functions
Var
Defined
SYNTAX Defined(E) where E is a CoCoA expression. SUMMARY check if an expression has been defined DESCRIPTION This function returns TRUE if E is defined, otherwise it returns false. Typically, it is used to check if a name has already been assigned. >EXAMPLE< Defined(MyVariable); FALSE ------------------------------- MyVariable := 3; Defined(MyVariable); TRUE -------------------------------
Deg
SYNTAX Deg(F:POLY or VECTOR):INT Deg(F:POLY or VECTOR,X:INDET):INT SUMMARY the degree of a polynomial or vector DESCRIPTION The first form of this function returns the (weighted) degree of F. The second form returns the (un-weighted) degree of the indeterminate X in F. In either case, if F is a vector, the maximum of the degrees of its components is returned. (For the degree of a ring or quotient object, see 'Multiplicity'.) >EXAMPLE< Use R ::= Q[x,y]; Deg(xy^2+y); 3 ------------------------------- Deg(xy^2+y,x); 1 ------------------------------- Use R ::= Q[x,y], Weights(2,3); Deg(xy^2+y); 8 ------------------------------- Deg(xy^2+y,x); 1 ------------------------------- Deg(Vector(x^2,xy^3+y,x^2-y^5)); 5 ------------------------------- Deg(Vector(x^2,xy^3+y,x^2-y^5),x); 2 ------------------------------ See also: MDeg Weights Modifier
DegLexMat, DegRevLexMat, LexMat, XelMat
SYNTAX DegLexMat(N:INTEGER):MAT DegRevLexMat(N:INTEGER):MAT LexMat(N:INTEGER):MAT XelMat(N:INTEGER):MAT SUMMARY matrices for std. term-orderings DESCRIPTION These functions return matrices defining standard term-orderings. >EXAMPLE< DegLexMat(3); Mat[ [1, 1, 1], [1, 0, 0], [0, 1, 0] ] ------------------------------- DegRevLexMat(3); Mat[ [1, 1, 1], [0, 0, -1], [0, -1, 0] ] ------------------------------- LexMat(3); Mat[ [1, 0, 0], [0, 1, 0], [0, 0, 1] ] ------------------------------- XelMat(3); Mat[ [0, 0, 1], [0, 1, 0], [1, 0, 0] ] ------------------------------- See also: Ord Orderings
Delete
SYNTAX Delete V_1, ..., V_n where each V_i is the identifier of a variable in the working memory. SUMMARY delete variables from the working memory DESCRIPTION This function removes variables from the working memory. It will not delete global variables. For more information about memory in CoCoA, see the chapter entitled "Memory Management." The command 'Memory()' lists the contents of the working memory. >EXAMPLE< Use R ::= Q[x,y,z]; X := Ideal(x,y); Y := 3; Use S ::= Q[a,b]; Z := a^2+b^2; Memory(); -- the contents of the working memory ["X", "Y", "Z"] ------------------------------- Delete X; Memory(); -- X has been deleted from the working memory ["It", "Y", "Z"] ------------------------------- See also: Clear Destroy Memory Memory Management
DensePoly
SYNTAX DensePoly(N:INT):POLY SUMMARY the sum of all power-products of a given degree DESCRIPTION This function returns the sum of all power-products of degree N. >EXAMPLE< Use R ::= Q[x,y]; DensePoly(3); x^3 + x^2y + xy^2 + y^3 ------------------------------- Use R::=Q[x,y],Weights(2,3); -- <--- NOTE DensePoly(1); 0 ------------------------------- DensePoly(6); x^3 + y^2 ------------------------------- See also: Randomize, Randomized
Der
SYNTAX Der(F,X:INDET):POLY where F is a polynomial or a rational function. SUMMARY the derivative of a rational function DESCRIPTION This function returns the derivative of F with respect to the indeterminate X. >EXAMPLE< Use R ::= Q[x,y]; Der(xy^2,x); y^2 ------------------------------- Define Jac(F) --> The Jacobian matrix for a polynomial. Return Mat([[Der(F,X) | X In Indets()]]); EndDefine; Jac(xy^2); Mat[ [y^2, 2xy] ] ------------------------------- Der(x/(x+y),x); y/(x^2 + 2xy + y^2) ------------------------------- See also: Jacobian
Describe
SYNTAX
Describe(E:OBJECT)
Describe E:OBJECT
SUMMARY
information about an expression
DESCRIPTION
This command gives information about the expression E.
>EXAMPLE<
Use R ::= Z/(32003)[t,x,y];
I := Ideal(t^3-x,t^4-y);
G := SyzOfGens(I);
Print I;
Ideal(t^3 - x, t^4 - y)
-------------------------------
Describe I;
Record[Type = IDEAL, Value = Record[Gens = [t^3 - x, t^4 - y],
SyzOfGens = Module([-t^4 + y, t^3 - x], [t^4x - xy, -t^3x + x^2])]]
-------------------------------
Describe Function('$cocoa/mat.Transposed');
Define Transposed(M)
If NOT(Type(M) = MAT) Then
Error("Transposed: argument must be a matrix");
EndIf;
Return(Mat[
J,
1..Len(M[1]),
TRUE,
[M[I][J]|I In 1..Len(M)]
]);
EndDefine;
-------------------------------
See also:
Other Help
Destroy
SYNTAX
Destroy R_1, ... , R_n
where each R_i is the identifier of a ring.
SUMMARY
delete rings
DESCRIPTION
This command clears all global variables bound to the listed rings.
Moreover, if R is a ring in the list and there are no variables in the
current memory dependent upon R, then the ring identified by R is
deleted; otherwise R is renamed with a name of the form 'R#N' where N
is an integer. This renamed ring is automatically removed as soon as
the last variable dependent upon it is deleted.
The command will not work if one of the listed rings is the current
ring.
For more information about memory in CoCoA, see the chapter entitled
"Memory Management."
>EXAMPLE<
Use R ::= Q[x,y,z];
X := 3;
I := Ideal(x,y); -- dependent on R
ENV.R.Y := 5; -- in global memory bound to R
Use S ::= Q[a,b];
Destroy R;
RingEnvs(); -- R#1 created to hold because of the ideal I
["Q", "Qt", "R#1", "S", "Z"]
-------------------------------
Memory(); -- ENV.R.Y was destroyed along with R
["I", "It", "X"]
-------------------------------
I; -- I was not destroyed
R#1 :: Ideal(x, y)
-------------------------------
I := 3; -- overwrite I; it is no longer dependent on a CoCoA ring
Describe Memory();
------------[Memory]-----------
I = 3
It = R#1 :: Ideal(x, y)
X = 3
-------------------------------
RingEnvs(); -- subtle point here: the variable 'It' is still dependent
-- on R#1
["Q", "Qt", "R#1", "S", "Z"]
-------------------------------
RingEnvs(); -- However, the previous command caused It to becomed a
-- string; hence, R#1 disappears.
["Q", "Qt", "S", "Z"]
-------------------------------
See also:
Clear
Delete
Memory Management
Det
SYNTAX Det(M:MAT) the resulting type depends on the entries of the matrix. SUMMARY the determinant of a matrix DESCRIPTION This function returns the determinant of the matrix M. The resulting type depends on the types of the entries of the matrix. >EXAMPLE< Use R ::= Q[x]; M := Mat([[x,x^2],[x,x^3]]); Det(M); x^4 - x^3 ------------------------------- Det(Mat([[1,2],[0,5]])); 5 ------------------------------- See also: Minors
Diff
SYNTAX Diff(L:LIST,M:LIST):LIST SUMMARY returns the difference between two lists DESCRIPTION This function returns the list obtained by removing all the elements of M from L. >EXAMPLE< L := [1,2,3,2,[2,3]]; M := [1,2]; Diff(L,M); [3, [2, 3]] ------------------------------- See also: Remove
Dim
SYNTAX
Dim(R:RING or TAGGED("Quotient")):INT
SUMMARY
the dimension of a ring or quotient object
DESCRIPTION
This function computes the dimension of R. The weights of the
indeterminates of the current ring must all be 1.
The coefficient ring must be a field.
>EXAMPLE<
Use R ::= Q[x,y,z];
Dim(R);
3
-------------------------------
Dim(R/Ideal(y^2-x,xz-y^3));
1
-------------------------------
Discriminant
SYNTAX Discriminant(F:POLY):POLY Discriminant(F:POLY, X:INDET):POLY SUMMARY the discriminant of a polynomial DESCRIPTION This function computes the discriminant of a polynomial F (with respect to a given indeterminate X, if the polynomial is multivariate). If the polynomial is univariate then there is no need to specify which indeterminate to use. The discriminant is defined to be the resultant of F and its derivative with respect to X. >EXAMPLE< Use R ::= Q[x,y]; Discriminant(x^2+3y^2, x); 12y^2 ------------------------------- Discriminant(x^2+3y^2, y); 36x^2 ------------------------------- Discriminant((x+1)^20+2); 54975581388800000000000000000000 ------------------------------- See also: Resultant
Distrib
SYNTAX Distrib(L:LIST):LIST SUMMARY the distribution of objects in a list DESCRIPTION For each object E of a list L, let N(E) be the number of times E occurs as a component of L. Then Distrib(L) returns the list whose components are [E,N(E)]. >EXAMPLE< Distrib(['b','a','b',4,4,[1,2]]); [["b", 2], ["a", 1], [4, 2], [[1, 2], 1]] ------------------------------- See also: Count
Div, Mod
SYNTAX Div(N:INT,D:INT):INT Mod(N:INT,D:INT):INT SUMMARY quotient and remainder for integers DESCRIPTION If N = Q*D + R, and 0 <= R < |D|, then Div(N,D) returns Q and Mod(N,D) returns R. NOTE: To perform the division algorithm on a polynomial or vector, use 'NR' (normal remainder) to find the remainder, or 'DivAlg' to get both the quotients and the remainder. To determine if a polynomial is in a given ideal or a vector is in a given module, use 'NF' or 'IsIn', and to find a representation in terms of the generators use 'GenRepr'. >EXAMPLE< Div(10,3); 3 ------------------------------- Mod(10,3); 1 ------------------------------- See also: DivAlg GenRepr NF NR
DivAlg
SYNTAX DivAlg(X:POLY,L:LIST of POLY):RECORD DivAlg(X:VECTOR,L:LIST of VECTOR):RECORD SUMMARY division algorithm DESCRIPTION This function performs the division algorithm on X with respect to L. It returns a record with two fields: 'Quotients' holding a list of polynomials, and 'Remainder' holding the remainder of X upon division by L. >EXAMPLE< Use R::= Q[x,y,z]; F := x^2y+xy^2+y^2; L := [xy-1,y^2-1]; DivAlg(F,[xy-1,y^2-1]); Record[Quotients = [x + y, 1], Remainder = x + y + 1] ------------------------------- D := It; D.Quotients; [x + y, 1] ------------------------------- D.Remainder; x + y + 1 ------------------------------- ScalarProduct(D.Quotients,L) + D.Remainder = F; TRUE ------------------------------- V := Vector(x^2+y^2+z^2,xyz); L := [Vector(x,y),Vector(y,z),Vector(z,x)]; DivAlg(V,L); Record[Quotients = [0, -z^2, yz], Remainder = Vector(x^2 + y^2 + z^2, z^3)] ------------------------------- See also: Div, Mod GenRepr NF NR
E_
SYNTAX E_(K:INT,N:INT or MODULE):VECTOR SUMMARY canonical vector DESCRIPTION If N is an integer, this function returns the K-th canonical vector of the free module of rank N over the current ring. If N is a module, it returns the K-th canonical vector of N. >EXAMPLE< Use R ::= Q[x,y]; E_(4,7); Vector(0, 0, 0, 1, 0, 0, 0) ------------------------------- M := Module([x^2,0,y^2],[x^3,x+y,y^3]); E_(2,M); Vector(0, 1, 0) -------------------------------
Elim
SYNTAX Elim(X:INDETS,M:IDEAL):IDEAL Elim(X:INDETS,M:MODULE):MODULE where X is an indeterminate or a list of indeterminates. SUMMARY eliminate variables DESCRIPTION This function returns the ideal or module obtained by eliminating the indeterminates X from M. The coefficient ring needs to be a field. As opposed to this function, there is also the *modifier*, 'Elim', used when constructing a ring (see 'Orderings' and 'Predefined Term-Orderings'). >EXAMPLE< Use R ::= Q[t,x,y,z]; Set Indentation; Elim(t,Ideal(t^15+t^6+t-x,t^5-y,t^3-z)); Ideal( -z^5 + y^3, -y^4 - yz^2 + xy - z^2, -xy^3z - y^2z^3 - xz^3 + x^2z - y^2 - y, -y^2z^4 - x^2y^3 - xy^2z^2 - yz^4 - x^2z^2 + x^3 - y^2z - 2yz - z, -y^3z^3 + xz^3 - y^3 - y^2) ------------------------------- Use R ::= Q[t,s,x,y,z,w]; t..x; [t, s, x] ------------------------------- Elim(t..x,Ideal(t-x^2zw,x^2-t,y^2t-w)); -- Note the use of t..x. Ideal(-zw^2 + w) ------------------------------- Use R ::= Q[t[1..2],x[1..4]]; I := Ideal(x[1]-t[1]^4,x[2]-t[1]^2t[2],x[3]-t[1]t[2]^3,x[4]-t[2]^4); t; [t[1], t[2]] ------------------------------- Elim(t,I); -- Note the use t. Ideal(x[3]^4 - x[1]x[4]^3, x[2]^4 - x[1]^2x[4]) ------------------------------- See also: Orderings Predefined Term-Orderings
EqSet
SYNTAX EqSet(L:LIST,M:LIST):BOOL SUMMARY checks if the set of elements in two lists are equal DESCRIPTION This function returns TRUE if Set(L) equals Set(M), otherwise it returns FALSE. >EXAMPLE< L := [1,2,2]; M := [2,1]; EqSet(L,M); TRUE ------------------------------- See also: Intersection, IntersectionList SubSet
Equals
SYNTAX Equals() SUMMARY returns a string of equal signs DESCRIPTION This function returns a string of equal signs: >EXAMPLE< Equals(); =============================== ------------------------------- See also: Dashes
EquiIsoDec
SYNTAX EquiIsoDec(I:IDEAL):LIST of IDEAL SUMMARY equidimensional isoradical decomposition DESCRIPTION This function computes an equimensional isoradical decomposition of I, i.e. a list of unmixed ideals I_1,...,I_k such that the radical of I is the intersection of the radicals of I_1,...,I_k. Redundancies are possible. NOTE: at the moment, this implementation works only if the coefficient ring is the rationals or has large enough characteristic. >EXAMPLE< Use R::=Q[x,y,z]; I := Intersection(Ideal(x-1,y-1,z-1),Ideal(x-2,y-2)^2,Ideal(x)^3); H := EquiIsoDec(I); H; [Ideal(x), Ideal(z - 1, y - 1, x - 1), Ideal(xy - y^2 - 2x + 2y, x^2 - y^2 - 4x + 4y, y^2z - y^2 - 4yz + 4y + 4z - 4, y^3 - 5y^2 + 8y - 4, x - 2)] ------------------------------- T := [Radical(J)|J In H]; S := IntersectionList(T); Radical(I) = S; TRUE ------------------------------- See also: Radical RadicalOfUnmixed
Error
SYNTAX
Error(S:STRING):ERROR
SUMMARY
return an error message
DESCRIPTION
This function returns an error labeled with the string S.
>EXAMPLE<
Define T(N)
If Type(N) <> INT Then Error("Argument must be an integer.") EndIf;
Return Mod(N,5);
EndDefine;
T(1/3);
-------------------------------
ERROR: Argument must be an integer.
CONTEXT: Error("Argument must be an integer.")
-------------------------------
T(7);
2
-------------------------------
See also:
Catch
GetErrMesg
Eval
SYNTAX Eval(E:OBJECT,L:LIST):OBJECT SUMMARY substitute numbers or polynomials for indeterminates DESCRIPTION This function substitutes the N-th element of L for the N-th indeterminate of the current ring for all N less than or equal to the minimum of the number of indeterminates of the current ring and the number of components of L. >EXAMPLE< Use Q[xy]; Eval(x^2+y,[2, 3]); 7 ------------------------------- Eval(x^2+y,[2]); y + 4 ------------------------------- F:=x(x-1)(x-2)y(y-1)(y-2)/36; P:=[1/2, -2/3]; Eval(F, P); -5/162 ------------------------------- Eval([x+y,x-y],[2,1]); [3, 1] ------------------------------- Eval([x+y,x-y],[x^2,y^2]); [x^2 + y^2, x^2 - y^2] ------------------------------- Eval([x+y,x-y],[y]); [2y, 0] ------------------------------- See also: Evaluation of Polynomials Image Subst Substitutions
EvalHilbertFn
SYNTAX
EvalHilbertFn(H:TAGGED("$cocoa/hp.Hilbert"),N:INT):INT
SUMMARY
evaluate the Hilbert function
DESCRIPTION
This function evaluates the Hilbert function H at N. If H is the
Hilbert function of a quotient R/I, then the value returned is the
same as that returned by 'Hilbert(R/I,N)' but time is saved since the
Hilbert function does not need to be recalculated at each call.
>EXAMPLE<
Use R ::= Q[w,x,y,z];
I := Ideal(z^2-xy,xz^2+w^3);
H := Hilbert(R/I);
H;
H(0) = 1
H(1) = 4
H(t) = 6t - 3 for t >= 2
-------------------------------
EvalHilbertFn(H,1);
4
-------------------------------
EvalHilbertFn(H,2);
9
-------------------------------
See also:
Hilbert
HilbertPoly
Fact
SYNTAX Fact(N:INT):INT where N is a non-negative integer. SUMMARY factorial function DESCRIPTION This function returns N factorial. >EXAMPLE< Fact(5); 120 ------------------------------- Fact(100); 9332621544394415268169923885626670049071596826438162146859 2963895217599993229915608941463976156518286253697920827223 758251185210916864000000000000000000000000 ------------------------------- See also: Bin
Factor
SYNTAX
Factor(F:POLY):LIST
SUMMARY
factor a polynomial
DESCRIPTION
This function factors a polynomial in its ring of definition.
Multivariate factorization is not yet supported over finite fields.
(For information about the algorithm, consult 'Pointers to the
Literature'.)
The function returns a list of the form [[F_1,N_1],...,[F_r,N_r]]
where F_1^N_1 ... F_r^N_r = F and the F_i are irreducible in the
current ring.
>EXAMPLE<
Use R ::= Q[x,y];
F := x^12 - 37x^11 + 608x^10 - 5852x^9 + 36642x^8 - 156786x^7 + 468752x^6
- 984128x^5 + 1437157x^4 - 1422337x^3 + 905880x^2 - 333900x + 54000;
Factor(F);
[[x - 2, 1], [x - 4, 1], [x - 6, 1], [x - 3, 2], [x - 5, 3], [x - 1, 4]]
---------------------------------
G := Product([W[1]^W[2] | W In It]); -- check solution
F = G;
TRUE
---------------------------------
Factor((8x^2+16x+8)/27); -- the "content" appears as a factor of degree 0;
-- it is not factorized into prime factors.
[[x + 1, 2], [8/27, 1]]
---------------------------------
F := (x+y)^2*(x^2y+y^2x+3);
F;
x^4y + 3x^3y^2 + 3x^2y^3 + xy^4 + 3x^2 + 6xy + 3y^2
-------------------------------
Factor(F); -- multivariate factorization
[[x^2y + xy^2 + 3, 1], [x + y, 2]]
-------------------------------
Use Z/(37)[x];
Factor(x^6-1);
[[x - 1, 1], [x + 1, 1], [x + 10, 1], [x + 11, 1], [x - 11, 1], [x - 10, 1]]
---------------------------------
Factor(2x^2-4); -- over a finite field the factors are made monic;
-- leading coeff appears as "content" if it is not 1.
[[x^2 - 2, 1], [2, 1]]
---------------------------------
Fields
SYNTAX Fields(P:RECORD):LIST SUMMARY list the fields of a record DESCRIPTION This function returns a list of all of the fields of the record P. >EXAMPLE< P := Record[ Name = 'David', Number = 3728852, Data = ['X','Y'] ]; Fields(P); ["Data", "Name", "Number"] ------------------------------- P.Data; ["X", "Y"] ------------------------------- See also: Introduction to Records Record
First
SYNTAX First(L:LIST):OBJECT First(L:LIST,N:INT):OBJECT SUMMARY the first N elements of a list DESCRIPTION In the first form, the function is the same as the function 'Head'; it returns the first element of the list L. In the second form, it returns the list of the first N elements of L. >EXAMPLE< L := [1,2,3,4,5]; First(L); 1 ------------------------------- First(L,3); [1, 2, 3] ------------------------------- See also: Head Last
FirstNonZero, FirstNonZeroPos
SYNTAX FirstNonZero(V:VECTOR):POLY FirstNonZeroPos(V:VECTOR):POLY SUMMARY the first non-zero entry in a vector DESCRIPTION The first function returns the first non-zero entry of V. The second function returns the index of the first non-zero entry of V. If either function is handed a zero vector then an error is signalled. >EXAMPLE< Use R ::= Q[x,y,z]; V := Vector(0,0,x^2+yz,0,z^2); FirstNonZero(V); x^2 + yz ------------------------------- FirstNonZeroPos(V); 3 ------------------------------- V[FirstNonZeroPos(V)]; x^2 + yz ------------------------------- See also: NonZero
Flatten
SYNTAX Flatten(L:LIST):LIST Flatten(L:LIST,N:INT):LIST SUMMARY flatten a list DESCRIPTION Components of lists may be lists themselves, i.e., lists may be nested. With one argument this function returns the list obtained from the list L by removing all nesting, bringing all elements "to the top level." With the optional second argument, N, nesting is removed down N levels. Thus, the elements of M := Flatten(L,1) are formed as follows: go through the elements of L one at a time; if an elements is not a list, add it to M; if an element is a list, add all of its elements to M. Recursively, Flatten(L,N) = Flatten(Flatten(L,N-1),1). For N large, depending on L, Flatten(L,N) gives the same result as Flatten(L). >EXAMPLE< Flatten([1,["a","b",[2,3,4],"c","d"],5,6]); [1, "a", "b", 2, 3, 4, "c", "d", 5, 6] ------------------------------- L := [1,2, [3,4], [5, [6,7,[8,9]]]]; Flatten(L,1); [1, 2, 3, 4, 5, [6, 7, [8, 9]]] ------------------------------- Flatten(It,1); [1, 2, 3, 4, 5, 6, 7, [8, 9]] ------------------------------- Flatten(L,2); -- same as in the previous line [1, 2, 3, 4, 5, 6, 7, [8, 9]] ------------------------------- Flatten(L,3); -- same as Flatten(L) [1, 2, 3, 4, 5, 6, 7, 8, 9] -------------------------------
FloatStr, MantissaAndExponent
SYNTAX FloatStr(X:RAT):STRING FloatStr(X:RAT, Prec:INT):STRING MantissaAndExponent(X:RAT, Prec:INT):RECORD SUMMARY convert rational number to a float string DESCRIPTION The first two functions convert a rational number X into a (decimal) floating point string. The optional second argument Prec says how many decimal digits to include in the mantissa; the default value is 10. Note that an exponent is always included; the only exception being the number zero which is converted to the string "0". The third function converts a rational number into a Record with components named Mantissa and Exponent. The value of the Exponent field is the unique integer E such that 1 <= X*10^E < 10, and the value of Mantissa is the nearest integer to (X*10^E)*10^(Prec-1). As an exception the case of X=0 always produces zero values for both components of the record. >EXAMPLE< FloatStr(2/3); -- last printed digit is rounded 6.666666667*10^(-1) ------------------------------- FloatStr(7^510); -- no arbitrary limit on exponent range 1.000000938*10^431 ------------------------------- FloatStr(1/81, 50); -- precision of mantissa specified by user 1.2345679012345679012345679012345679012345679012346*10^(-2) ------------------------------- FloatStr(1/2); -- trailing zeroes are not suppressed 5.000000000*10^(-1) ------------------------------- MantissaAndExponent(1/2,3); -- 1/2 = 5.00*10^(-1) Record[Exponent = -1, Mantissa = 500] ------------------------------- MantissaAndExponent(0.9999, 3); -- 0.9999 rounds up to give 1.00 Record[Exponent = 0, Mantissa = 100] -------------------------------
For
SYNTAX
For I := N_1 To N_2 Do C EndFor
For I := N_1 To N_2 Step D Do C EndFor
where I is a dummy variable, N_1, N_2, and D are integer expressions,
and C is a sequence of commands.
SUMMARY
loop command
DESCRIPTION
In the first form, the variable I is assigned the values N_1, N_1+1,
..., N_2 in succession. After each assignment, the command sequence C
is executed. The second form is the same, except that I is assigned
the values N_1, N_1+D, N_1+2D, etc. until the greatest value less than
or equal to N_2 is reached. If N_2 < N_1, then C is not executed.
Note: Large values for N_1, N_2, or D are not permitted; typically
they should lie in the range about -10^9 to +10^9.
Note: Don't forget the capitalization in the word 'To'.
>EXAMPLE<
For N := 1 To 5 Do Print(2^N,' ') EndFor;
2 4 8 16 32
-------------------------------
For N := 1 To 20 Step 3 Do Print(N,' ') EndFor;
1 4 7 10 13 16 19
-------------------------------
For N := 10 To 1 Step -2 Do Print(N,' ') EndFor;
10 8 6 4 2
-------------------------------
For N := 5 To 3 Do Print(N,' ') EndFor; -- no output
>EXAMPLE<
Loops can be nested.
Define Sort(Var(L))
For I := 1 To Len(L)-1 Do
M := I;
For J := I+1 To Len(L) Do
If L[J] < L[M] Then M := J EndIf;
EndFor;
If M <> I Then
C := L[M];
L[M] := L[I];
L[I] := C
EndIf
EndFor
EndDefine;
M := [5,3,1,4,2];
Sort(M);
M;
[1, 2, 3, 4, 5]
-------------------------------
(Note that 'Var(L)' is used so that the function can change the value
of the variable referenced by L. See 'Var'.)
See also:
Foreach
Repeat
While
Foreach
SYNTAX Foreach X In L Do C EndForeach where X is a dummy variable, L is a list, and C is a sequence of commands. SUMMARY loop command DESCRIPTION The dummy variable X is assigned the value of each component of L in turn. After each assignment the command sequence C is executed. Note: don't forget to capitalize 'In'. >EXAMPLE< Foreach N In 1..10 Do -- Note: 1..10 gives the list [1,...,10]. Print(N^2,' '); EndForeach; 1 4 9 16 25 36 49 64 81 100 ------------------------------- Use R ::= Q[x,y,z]; F := x^2y + 3y^2z - z^3; J := [Der(F,X) | X In Indets()]; -- the Jacobian for F J; [2xy, x^2 + 6yz, 3y^2 - 3z^2] ------------------------------- Foreach X In J Do -- square each component of the Jacobian PrintLn(X^2); EndForeach; 4x^2y^2 x^4 + 12x^2yz + 36y^2z^2 9y^4 - 18y^2z^2 + 9z^4 ------------------------------- See also: For Repeat While
Format
SYNTAX
Format(E:OBJECT,N:INT):STRING
SUMMARY
convert object to formatted string
DESCRIPTION
Like Sprint, this function converts the value of E into a string. If
the string has fewer than N characters, then spaces are added to the
front to make the length N.
>EXAMPLE<
L := [1,2,3];
M := Format(L,20);
M;
[1, 2, 3]
-------------------------------
Type(L);
LIST
-------------------------------
Type(M);
STRING
-------------------------------
Format(L,2); -- 'Format' does not truncate
[1, 2, 3]
-------------------------------
See also:
IO.SprintTrunc
Latex
Sprint
Fraction
SYNTAX Fraction(E:OBJECT,F:OBJECT) SUMMARY returns the quotient of its arguments DESCRIPTION This function returns E/F provided the quotient is defined (see 'Algebraic Operators'). >EXAMPLE< Use R ::= Q[x,y]; Fraction(2,3); 2/3 ------------------------------- Fraction(2,4); 1/2 ------------------------------- Fraction(x,x+y); x/(x + y) ------------------------------- Fraction(5%11,6%11); 10 % 11 ------------------------------- See also: Algebraic Operators
Function
SYNTAX
Function(S:STRING):FUNCTION
Function(P:STRING,S:STRING):FUNCTION
SUMMARY
return a function
DESCRIPTION
This function returns the function---user-defined or built
in---identified by the string S. In the second form, one first
provides the name of the package, then the name of the function. (An
alternative is the syntax 'Function(P.S)'.
One may use 'Function' to assign a function to a variable which can
then be executed via the function 'Call':
>EXAMPLE<
F := Function("Deg");
F;
Deg(...)
-------------------------------
Type(F);
FUNCTION
-------------------------------
Call(F,x+y^2);
2
-------------------------------
-- the Call-statement here is equivalent to:
Deg(x+y^2);
2
-------------------------------
Function('Insert');
Insert(L,I,O)
-------------------------------
Function('$cocoa/list','Insert'); -- or 'Function('$cocoa/list.Insert')'
Insert(L,I,O)
-------------------------------
See also:
Call
Functions
Functions
SYNTAX
Functions(S:STRING):LIST of FUNCTION
SUMMARY
list the functions of a package
DESCRIPTION
This function returns a list of functions defined in the package
identified by S. (The function 'Packages' lists the packages currently
loaded into memory.)
>EXAMPLE<
Functions('$cocoa/binrepr');
[About(), Man(), Initialize(), PolyBinRepr_xi(P), PolyBinRepr_xii(P),
BinExp(...), EvalBinExp(BE,Up,Down), Aux_BinExp(H,N), Tagged(X,T),
Print_Bin(B), Print_BinExp(BE), Print_BinRepr(BR), PkgName()]
-------------------------------
L:= It;
Describe L[5];
DEFINE BinExp(...)
IF Shape(ARGV) = [
INT,
INT] THEN
Return($cocoa/binrepr.Aux_BinExp(ARGV[1],ARGV[2]))
ELSIF Shape(ARGV) = [
INT,
INT,
INT,
INT] THEN
Return(EvalBinExp($cocoa/binrepr.Aux_BinExp(ARGV[1],ARGV[2]),ARGV[3],ARGV[4]))
ELSE
Error(ERR.BAD_PARAMS,"(BinExp arguments must be 2 or 4 INT)")
END;
END
-------------------------------
GB.Complete
SYNTAX GB.Complete(M:IDEAL or MODULE):NULL SUMMARY Complete an interactive Groebner-type calculation DESCRIPTION This function completes a calculation started in the Interactive Groebner Framework. For explanations and examples, see 'The Interactive Groebner Framework'. See also: The Interactive Groebner Framework
GB.GetBettiMatrix
SYNTAX
GB.GetBettiMatrix(M:IDEAL or MODULE):TAGGED("$cocoa/io.Matrix")
SUMMARY
returns the Betti matrix computed so far
DESCRIPTION
This function, if used after executing 'Res(M)', prints the Betti
matrix for M. Within the Interactive Groebner Framework, in which
resolutions may be computed one step at a time, the function returns
the Betti matrix for the part of the resolution computed so far. See
'GB.GetRes' for an example.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
I := Ideal(x^2-yt,xy-zt,xy);
Res(I);
0 --> R^2(-5) --> R^4(-4) --> R^3(-2)
-------------------------------
GB.GetBettiMatrix(I);
--------------
--------------
0 0 0
0 0 3
0 0 0
0 4 0
2 0 0
--------------
-------------------------------
See also:
GB.GetRes
GB.GetResLen
Res
The Interactive Groebner Framework
GB.GetNthSyz
SYNTAX GB.GetNthSyz(M:IDEAL or MODULE,N:INT):MODULE SUMMARY returns the part of the Nth syzygy module computed so far DESCRIPTION This function, if used after executing 'Res(M)', returns the Nth syzygy module for M. Within the Interactive Groebner Framework, in which resolutions may be computed one step at a time, the function returns the part of the Nth syzygy module computed so far. In contrast, the function 'Syz' always determines the complete syzygy module even from within the Interactive Groebner Framework. >EXAMPLE< Use R ::= Q[t,x,y,z]; I := Ideal(x^2-yt,xy-zt,xy); GB.Start_Res(I); GB.Step(I); GB.GetNthSyz(I,1); GB.GetNthSyz(I,2); Module([0]) ------------------------------- Module([0]) ------------------------------- GB.Step(I); GB.GetNthSyz(I,1); GB.GetNthSyz(I,2); Module([0, 0]) ------------------------------- Module([0]) ------------------------------- GB.Steps(I,5); GB.GetNthSyz(I,1); GB.GetNthSyz(I,2); Module([-xz, -y^2, yz]) ------------------------------- Module([0]) ------------------------------- GB.Complete(I); GB.GetNthSyz(I,1); GB.GetNthSyz(I,2); Module([-xz, -y^2, yz], [tz, xy, 0], [0, -x^2 + ty, -tz], [-x^2 + ty, 0, xy]) ------------------------------- Module([-x, -y, 0, z], [-t, -x, -y, 0]) ------------------------------- See also: GB.GetNthSyzShifts GB.GetRes Res Syz The Interactive Groebner Framework
GB.GetNthSyzShifts
SYNTAX
GB.GetNthSyzShifts(M:IDEAL or MODULE,N:INT):TAGGED("shifts")
SUMMARY
shifts of the Nth syzygy module computed so far
DESCRIPTION
This function, if used after executing 'Res(M)', returns the shifts
for the Nth syzygy module for M. Within the Interactive Groebner
Framework, in which resolutions may be computed one step at a time,
the function returns shifts of the part of the Nth syzygy module
computed so far.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
I := Ideal(x^2-yt,xy-zt,xy);
GB.Start_Res(I);
GB.Steps(I,6);
GB.GetNthSyzShifts(I,2);
Shifts([x^2yz])
-------------------------------
GB.Complete(I);
GB.GetNthSyzShifts(I,2);
Shifts([x^2yz, txyz, tx^2z, x^3y])
-------------------------------
J := Ideal(t,x)^3;
Res(J);
0 --> R^3(-4) --> R^4(-3)
-------------------------------
GB.GetNthSyzShifts(J,1);
Shifts([x^3, tx^2, t^2x, t^3])
-------------------------------
GB.GetNthSyzShifts(J,2);
Shifts([tx^3, t^2x^2, t^3x])
-------------------------------
SS := It;
SS[1];
tx^3
-------------------------------
See also:
GB.GetNthSyz
GB.GetRes
Res
Shifts
The Interactive Groebner Framework
GB.GetRes
SYNTAX
GB.GetRes(M:IDEAL or MODULE):TAGGED("$cocoa/gb.Res")
SUMMARY
returns the resolution computed so far
DESCRIPTION
This function returns the part of the resolution of M computed so
far. It does not compute the resolution of M as does 'Res', rather,
'GB.GetRes' is intended primarily to be used within the Interactive
Groebner Framework. Thus, for example, 'GB.GetRes' may be used to
examine the resolution as it is computed, one step at a time.
>EXAMPLE<
Use R ::= Q[txyz];
I := Ideal(x^2-yt,xy-zt,xy);
GB.Start_Res(I); -- start Interactive Groebner Framework
GB.Step(I); -- take one step in calculation of resolution
GB.GetRes(I); -- the resolution so far
0 --> R(-2)
-------------------------------
GB.Step(I); -- one more step
GB.GetResLen(I); -- the computed resolution still has length 1
1
-------------------------------
GB.GetBettiMatrix(I); -- the Betti Matrix so far
----
----
0
2
----
-------------------------------
GB.GetRes(I);
0 --> R^2(-2)
-------------------------------
GB.Steps(I,5); -- five more steps
GB.GetRes(I);
0 --> R(-4) --> R^3(-2)
-------------------------------
GB.Complete(I); -- complete the calculation
GB.GetResLen(I);
3
-------------------------------
GB.GetBettiMatrix(I);
--------------
--------------
0 0 0
0 0 3
0 0 0
0 4 0
2 0 0
--------------
-------------------------------
GB.GetRes(I);
0 --> R^2(-5) --> R^4(-4) --> R^3(-2)
-------------------------------
See also:
GB.GetBettiMatrix
GB.GetResLen
Res
The Interactive Groebner Framework
GB.GetResLen
SYNTAX GB.GetResLen(M:IDEAL or MODULE):INT SUMMARY returns the length of the resolution computed so far DESCRIPTION This function, if used after executing 'Res(M)', prints the length of the resolution for M. Within the Interactive Groebner Framework, in which resolutions may be computed one step at a time, the function returns the length of the part of the resolution computed so far. See 'GB.GetRes' for an example. >EXAMPLE< Use R ::= Q[t,x,y,z]; I := Ideal(x^2-yt,xy-zt,xy); Res(I); 0 --> R^2(-5) --> R^4(-4) --> R^3(-2) ------------------------------- GB.GetResLen(I); 3 ------------------------------- See also: GB.GetBettiMatrix GB.GetRes Res The Interactive Groebner Framework
GB.ResReport
SYNTAX GB.ResReport(M:IDEAL or MODULE):NULL SUMMARY status of an interactive resolution calculation DESCRIPTION This function reports statistics about the current status of a resolution computation begun in the Interactive Groebner Framework. For explanations and examples, see 'The Interactive Groebner Framework'. See also: GB.Stats The Interactive Groebner Framework
GB.Start_GBasis
SYNTAX GB.Start_GBasis(M:IDEAL or MODULE):NULL SUMMARY start interactive Groebner basis computation DESCRIPTION This command starts the Interactive Groebner Framework for calculating a Groebner basis for M. For explanations and examples, see 'The Interactive Groebner Framework'. See also: The Interactive Groebner Framework
GB.Start_MinGens
SYNTAX GB.Start_MinGens(M:IDEAL or MODULE):NULL SUMMARY start interactive minimal generator calculation DESCRIPTION This command starts the Interactive Groebner Framework for calculating minimal generators for M. For explanations and examples, see 'The Interactive Groebner Framework'. See also: The Interactive Groebner Framework
GB.Start_MinSyzMinGens
SYNTAX GB.Start_MinSyzMinGens: COMMAND ELIMINATED SUMMARY start interactive calc., min. syzs. of min. gens. DESCRIPTION The GB.Start_MinSyzMinGens command has been removed. See also: The Interactive Groebner Framework
GB.Start_Res
SYNTAX GB.Start_Res(M:IDEAL or MODULE):NULL SUMMARY start interactive resolution computation DESCRIPTION This command starts the Interactive Groebner Framework for calculating a resolution for M. For explanations and examples, see 'The Interactive Groebner Framework'. See also: The Interactive Groebner Framework
GB.Start_Syz
SYNTAX GB.Start_Syz(M:IDEAL or MODULE):NULL SUMMARY start interactive syzygy computation DESCRIPTION This command starts the Interactive Groebner Framework for calculating syzygies for M. For explanations and examples, see 'The Interactive Groebner Framework'. See also: The Interactive Groebner Framework
GB.Start_SyzMinGens
SYNTAX GB.Start_SyzMinGens: COMMAND ELIMINATED SUMMARY start interactive calc. of syzygies of min. gens. DESCRIPTION The GB.Start_SyzMinGens command has been removed. See also: The Interactive Groebner Framework
GB.Stats
SYNTAX GB.Stats(M:IDEAL or MODULE):NULL SUMMARY status of an interactive Groebner-type calculation DESCRIPTION This function displays information about the current status of a calculation started in the Interactive Groebner Framework. For explanations and examples, see 'The Interactive Groebner Framework'. See also: GB.ResReport The Interactive Groebner Framework
GB.Step, GB.Steps
SYNTAX GB.Step(M:IDEAL or MODULE):NULL GB.Steps(M:IDEAL or MODULE, N:INT):NULL SUMMARY take steps in an interactive Groebner-type calculation DESCRIPTION The first function performs one step in a calculation started in the Interactive Groebner Framework. The second, performs N steps. For explanations and examples, see 'The Interactive Groebner Framework'. See also: The Interactive Groebner Framework
GBasis
SYNTAX
GBasis(M:IDEAL, MODULE, or TAGGED("Quotient")):LIST
SUMMARY
calculate a Groebner basis
DESCRIPTION
If M is an ideal or module, this function returns a list whose
components form a Groebner basis for M with respect to the
term-ordering of the current ring. If M is a quotient of the current
ring by an ideal I or of a free module by a submodule N, then the
Groebner basis for M is defined to be that of I or N, respectively.
If M is a variable, then the result is stored in M for
later use. It can be retrieved as M.GBasis and can also be seen using
the command 'Describe'.
For a reduced Groebner basis, use the command 'ReducedGBasis'.
The coefficient ring must be a field.
>EXAMPLE<
Use R ::= Q[t,x,y];
I := Ideal(t^3-x,t^4-y);
Describe I;
Record[Type = IDEAL, Value = Record[Gens = [t^3 - x, t^4 - y]]]
-------------------------------
GBasis(I);
[t^3 - x, -tx + y, t^2y - x^2, x^3 - ty^2]
-------------------------------
Describe(I); -- the Groebner basis has been stored in I
Record[Type = IDEAL, Value = Record[Gens = [t^3 - x, t^4 - y], GBasis
= [t^3 - x, -tx + y, t^2y - x^2, x^3 - ty^2]]]
-------------------------------
I.GBasis;
[t^3 - x, -tx + y, t^2y - x^2, x^3 - ty^2]
-------------------------------
For fine control and monitoring of Groebner basis calculations, see
'The Interactive Groebner Framework' and 'Introduction to Panels'.
See also:
Introduction to Groebner Bases in CoCoA
GBM, HGBM
SYNTAX GBM(L:LIST):IDEAL HGBM(L:LIST):IDEAL SUMMARY intersection of ideals for zero-dimensional schemes DESCRIPTION These functions compute the intersection of ideals corresponding to zero-dimensional schemes: GBM is for affine schemes, and HGBM for projective schemes. The list L must be a list of ideals. The function 'IntersectionList' should be used for computing the intersection of a collection of general ideals. The name GBM comes from the name of the algorithm used: Generalized Buchberger-Moeller. The prefix H comes from Homogeneous since ideals of projective schemes are necessarily homogeneous. >EXAMPLE< Use Q[x,y,z]; I1:=IdealOfPoints([[1,2,1], [0,1,0]]); -- a simple affine scheme I2:=IdealOfPoints([[1,1,1], [2,0,1]])^2; -- another affine scheme GBM([I1,I2]); -- intersect the ideals Ideal(xz + yz - z^2 - x - y + 1, z^3 - 2z^2 + z, yz^2 - 2yz - z^2 + y + 2z - 1, y^2z - y^2 - yz + y, xy^2 + y^3 - 2x^2 - 5xy - 5y^2 + 2z^2 + 8x + 10y - 4z - 6, x^2y - y^3 + 2x^2 + 2xy + 4y^2 - 3z^2 - 8x - 8y + 6z + 5, x^3 + y^3 - 7x^2 - 5xy - 4y^2 + 5z^2 + 16x + 10y - 10z - 7, y^4 - 2y^3 - 4x^2 - 8xy - 3y^2 + 4z^2 + 16x + 16y - 8z - 12) ------------------------------- Use Q[x[0..2]]; I1:=IdealOfProjectivePoints([[1,2,1], [0,1,0]]); -- simple projective scheme I2:=IdealOfProjectivePoints([[1,1,1], [2,0,1]])^2; -- another projective scheme HGBM([I1,I2]); -- intersect the ideals Ideal(x[0]^3 - x[0]x[1]^2 - 5x[0]^2x[2] + x[1]^2x[2] + 8x[0]x[2]^2 - 4x[2]^3, x[0]^2x[1] + x[0]x[1]^2 - 3x[0]x[1]x[2] - x[1]^2x[2] + 2x[1]x[2]^2, x[0]x[1]^3 - 2x[0]^2x[2]^2 - 5x[0]x[1]x[2]^2 - 4x[1]^2x[2]^2 + 8x[0]x[2]^3 + 10x[1]x[2]^3 - 8x[2]^4, x[0]x[1]^2x[2] + x[1]^3x[2] - 2x[0]^2x[2]^2 - 5x[0]x[1]x[2]^2 - 5x[1]^2x[2]^2 + 8x[0]x[2]^3 + 10x[1]x[2]^3 - 8x[2]^4, x[1]^4x[2] - 2x[1]^3x[2]^2 - 4x[0]^2x[2]^3 - 8x[0]x[1]x[2]^3 - 3x[1]^2x[2]^3 + 16x[0]x[2]^4 + 16x[1]x[2]^4 - 16x[2]^5) ------------------------------- See also: Finite Point Sets: Buchberger-Moeller IdealAndSeparatorsOfPoints IdealAndSeparatorsOfProjectivePoints IdealOfPoints IdealOfProjectivePoints
GCD, LCM
SYNTAX GCD (F_1:INT,...,F_n:INT):INT GCD (L:LIST of INT):INT LCM (F_1:INT,...,F_n:INT):INT LCM (L:LIST of INT):INT GCD(F_1:POLY,...,F_n:POLY):POLY GCD (L:LIST of POLY):POLY LCM(F_1:POLY,...,F_n:POLY):POLY LCM (L:LIST of POLY):POLY SUMMARY greatest common divisor, least common multiple DESCRIPTION These functions return the greatest common divisor and least common multiple, respectively, of F_1,...,F_n or of the elements in the list L. For the calculation of the GCDs and LCMs of polynomials, the coefficient ring must be a field. >EXAMPLE< Use R ::= Q[x,y]; F := x^2-y^2; G := (x+y)^3; GCD(F,G); x + y ------------------------------- LCM(F,G); 1/4x^4 + 1/2x^3y - 1/2xy^3 - 1/4y^4 ------------------------------- 4It = (x+y)^3(x-y); TRUE ------------------------------- GCD(3*4,3*8,6*16); 12 ------------------------------- GCD([3*4,3*8,6*16]); 12 ------------------------------- See also: Div, Mod
GenericPoints
SYNTAX GenericPoints(NumPoints:INT):LIST GenericPoints(NumPoints:INT,RandomRange:INT):LIST SUMMARY random projective points DESCRIPTION 'GenericPoints' returns a list of NumPoints generic projective points with integer coordinates; it is not guaranteed that these points are distinct. RandomRange specifies the largest value any coordinate may take. If the second argument is omitted, the largest value possible is 100 (or P-1 where P is the characteristic of the coefficient ring). >EXAMPLE< Use R ::= Q[x,y];GenericPoints(7); [[1, 0], [0, 1], [1, 1], [12, 59], [6, 63], [12, 80], [17, 63]] ------------------------------- GenericPoints(7,500); [[1, 0], [0, 1], [1, 1], [220, 162], [206, 452], [98, 106], [403, 449]] ------------------------------- Use R ::= Z/(5)[x,y,z]; GenericPoints(7); [[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1], [2, 1, 1], [2, 2, 4], [3, 1, 3]] ------------------------------- GenericPoints(7,500); [[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1], [1, 4, 2], [1, 3, 2], [2, 3, 3]] -------------------------------
GenRepr
SYNTAX
GenRepr(X:POLY,I:IDEAL):LIST of POLY
GenRepr(X:VECTOR,I:MODULE):LIST of POLY
SUMMARY
representation in terms of generators
DESCRIPTION
This function returns a list giving a representation of X in terms of
generators for I. Let the generators for I be [G_1,...,G_t]. If X is
in I, then 'GenRepr' will return a list [F_1,...,F_t] such that
X = F_1*G_1 + ... + F_t*G_t.
If X is not in I, then 'GenRepr' returns the empty list, [].
>EXAMPLE<
Use R ::= Q[xy];
I := Ideal(x+y^2,x^2-xy);
GenRepr(x^3-x^2y-y^3-xy,I);
[-y, x]
-------------------------------
-y I.Gens[1] + x I.Gens[2];
x^3 - x^2y - y^3 - xy
-------------------------------
GenRepr(x+y,I);
[ ]
-------------------------------
x+y IsIn I; -- the empty list was returned above since x+y is not in I
FALSE
-------------------------------
V1:= Vector(x,y,y^2); V2:= Vector(x-y,0,x^2);
X := x^2 V1 - y^2 V2;
M := Module(V1,V2);
GenRepr(X,M);
[x^2, -y^2]
-------------------------------
See also:
DivAlg
IsIn
NF
Gens
SYNTAX Gens(I:IDEAL):LIST Gens(M:MODULE):LIST SUMMARY list of generators of an ideal DESCRIPTION This function returns a list of polynomials which generate the ideal I or the module M. The list is not necessarily minimal. Note that I.Gens and M.Gens will give the same lists of generators. The advantage of 'Gens' is that its argument can be any expression evaluating to an ideal or module. >EXAMPLE< Use R ::= Q[x,y]; I := Ideal(y^2-x^3,xy); Gens(I); [-x^3 + y^2, xy] ------------------------------- I.Gens; [-x^3 + y^2, xy] ------------------------------- Gens(I^2); [x^6 - 2x^3y^2 + y^4, -x^4y + xy^3, x^2y^2] ------------------------------- See also: Minimalize Minimalized
Get
SYNTAX
Get(D:DEVICE,N:INT):LIST of INT
SUMMARY
read characters from a device
DESCRIPTION
This function reads N characters from D and returns the list of
their ascii codes.
>EXAMPLE<
D := OpenIFile('io.pkg'); -- open the file 'io.pkg'
Get(D,10); -- get the first 10 characters
[45, 45, 32, 105, 111, 100, 101, 118, 46, 112]
-------------------------------
Ascii(It); convert the ascii code to characters
-- iodev.p
-------------------------------
Ascii(Get(D,10)); -- get the next 10 characters and convert
kg : 0.1 :
-------------------------------
Close(D);
See also:
Introduction to IO
OpenIFile, OpenOFile
OpenIString, OpenOString
GetErrMesg
SYNTAX GetErrMesg(E:ERROR):STRING SUMMARY returns the message associated with an error DESCRIPTION This function returns the error message associated with an error. >EXAMPLE< Str := GetErrMesg(1/0); PrintLn(Str); Division by zero ------------------------------- See also: Catch Error
GlobalMemory
SYNTAX
GlobalMemory():TAGGED("Memory")
SUMMARY
contents of global memory
DESCRIPTION
This function prints the contents of the global memory which are not
bound to rings: variables prefixed by 'MEMORY' but not by
'MEMORY.ENV'. Untagging the value returned by 'GlobalMemory' gives a
list of strings which are identifiers for the global variables. The
command 'Fields(MEMORY)' gives the same set of strings.
For more information about memory in CoCoA, see the chapter entitled
"Memory Management."
>EXAMPLE<
Use R ::= Q[x,y,z];
A := 3;
ENV.R.B := 7;
MEMORY.C := 6;
GlobalMemory();
["C", "DEV", "ENV", "ERR", "PKG"]
-------------------------------
MEMORY.ENV; -- the record holding the rings defined during the
-- CoCoA session
Record[Q = Q, Qt = Q[t], R = Q[x,y,z], Z = Z]
-------------------------------
Memory(); -- the working memory
["A", "It"]
-------------------------------
Memory(R); -- the global variables bound to the ring R
["B"]
-------------------------------
See also:
Memory
Memory Management
H.Browse
SYNTAX H.Browse():NULL; H.Browse(N:INT):NULL SUMMARY browse the online help system DESCRIPTION This function browses the online help system. Without an argument, it displays the next section of the online manual or the next command. With integer argument N, it skips ahead (or skips back, if N is negative) N sections of the manual or N commands. See also: H.Commands ?, Man
H.Commands
SYNTAX
H.Commands():NULL;
H.Commands(S:STRING):NULL
SUMMARY
list commands according to type
DESCRIPTION
This function prints a list of commands associated with the string S.
For example, 'H.Commands('poly')' will list all documented commands
having to do with polynomials. Unlike '?', this function searches
only the list of commands (not both the list of commands and online
manual, both of which are a part of the online help system). Also,
unlike '?', this function does not try to use the string S to identify
a unique function. Instead it looks for all functions whose "type"
is S, i.e., that are somehow related with the search string S. The
types are often the names of data types in CoCoA. A complete list of
these types, along with additional information, can be found by
entering 'H.Commands()', without any argument.
After a command name is found, complete information on the command can
be obtained using '?'. The function 'H.Syntax' prints just the syntax
for the command.
Note: entering 'H.Commands('')' will produce a complete list of the
documented commands.
See also:
?, Man
H.Syntax
H.Man
SYNTAX H.Man():NULL H.Man(S:STRING):NULL H.Man(S:STRING,0):NULL where 0 is the integer 0. SUMMARY search online help system (see '?, Man') DESCRIPTION This function is synonymous with 'Man', which performs the same task as '?'. See '?, Man' for more information. See also: H.Commands H.Syntax ?, Man
H.OutCommands
SYNTAX
H.OutCommands(S:STRING):NULL
H.OutCommands(S:STRING,A:INT):NULL
H.OutCommands(S:STRING,A:INT,B:INT):NULL
SUMMARY
print command descriptions to a file
DESCRIPTION
The function prints the online descriptions of commands to the text
file named S. Warning: if a file named S already exists, it is
appended to. The first form prints all of the command descriptions,
the second prints only the command with number A, and the last prints
commands with numbers A to B. The total number of commands is given
by Len(MEMORY.Doc.Commands). The name of the command with number I is
MEMORY.Doc.Commands[I].Title. Entering 'H.Commands('')' will list the
documented commands, in order.
>EXAMPLE<
H.OutCommands('CommandFile',1,10);
To print sections of the online manual, use the function 'H.OutManual'.
See also:
H.OutManual
H.OutManual
SYNTAX
H.OutManual(S:STRING):NULL
H.OutManual(S:STRING,P:INT):NULL
H.OutManual(S:STRING,P:INT,C:INT):NULL
H.OutManual(S:STRING,P:INT,C:INT,S:INT):NULL
SUMMARY
print the online manual to a file
DESCRIPTION
This function prints sections of the manual to a text file named S.
Warning: if a file named S already exists, it is appended to. The
first form prints the entire manual to a file. The others are used to
print part P, chapter C, section S. Recall that the online help
consists of a manual *and* a list of commands. To print out the
commands, use 'H.OutCommands'.
>EXAMPLE<
H.OutManual('part1.chp2',1,2);
See also:
H.OutCommands
H.Toc
H.SetMore, H.UnSetMore
SYNTAX H.SetMore(N:INT):NULL H.SetMore():NULL H.UnSetMore():NULL SUMMARY more-device for online help DESCRIPTION The purpose of these functions is to turn on and off filtering of the online help system through the function 'More'. When the online help system filters through 'More' any output from online help is stored in a 'MoreDevice' then printed to the screen, N lines at a time. The number N is stored in the global variable MEMORY.MoreCount and may be set directly by the user with the command 'MEMORY.MoreCount := X' where X is an integer. The idea is to keep the output from scrolling off of the screen. See 'More' for more information. The function 'H.SetMore' turns on filtering through 'More'; and if the optional argument N is supplied, it sets MEMORY.MoreCount to N. The function 'H.UnSetMore' turns off filtering through 'More', without affecting MEMORY.MoreCount. See also: More
H.Syntax
SYNTAX
H.Syntax():NULL
H.Syntax(S:STRING):NULL
SUMMARY
display the syntax of a command
DESCRIPTION
The first form of the command, with no arguments, just prints this
message. The second form looks for a command with associated keywords
containing S as a substring. If S is exactly the keyword of a
command or if S is the substring of a keyword of only one command,
then the syntax for that command is displayed. (The command 'H.Browse'
can then be called to display additional information.) Otherwise,
H.Syntax(S) lists the names all commands with associated keywords
containing S as a substring. Note: the search is case insensitive.
>EXAMPLE<
H.Syntax('dense');
DensePoly(N:INT):POLY
Description: the sum of all power-products of a given degree
--> 'H.Browse();' for more information. <--
-------------------------------
H.Tips
SYNTAX
H.Tips():NULL
SUMMARY
quick tips for using online help
DESCRIPTION
This function prints advice on using CoCoA's online help system
effectively.
>EXAMPLE<
H.Tips();
============ Quick Tips for Using Online Help =============
Here are some tips for using the online help system:
1. Searches are case insensitive and your search string need only be a
substring of a keyword to make a match. Thus, for instance, to find
the section of the manual entitled "Commands and Functions for
Polynomials", it is enough to type: '?for poly'.
---> Output suppressed <---
See also:
Online Help
H.Toc
SYNTAX
H.Toc():NULL;
H.Toc(P:INT):NULL
H.Toc(P:INT,C:INT):NULL
H.Toc("all"):NULL
SUMMARY
display the Table Of Contents of the online manual
DESCRIPTION
The first form of this function, with no arguments, lists the titles
of the parts and chapters of the manual. The second prints the table
of contents for part P. The third prints the table of contents for
part P, chapter C. The last form, with the string "all" as argument,
prints the entire table of contents.
The contents of each section can be read online by giving enough of
its title as an argument to '?'.
See also:
?, Man
H.Tutorial
SYNTAX H.Tutorial():NULL SUMMARY run the CoCoA tutorial DESCRIPTION The CoCoA tutorial is part of the online manual. This function displays the first section of the tutorial. The following sections can then be browsed using 'H.Browse'. See also: H.Browse
Head
SYNTAX Head(L:LIST):OBJECT SUMMARY the first element of a list DESCRIPTION This function returns the first element of the list L. >EXAMPLE< Head([3,2,1]); 3 ------------------------------- See also: First Last Tail
Help
SYNTAX
Help(S:STRING):NULL
SUMMARY
extend online help for user-defined functions
DESCRIPTION
This command is used for extending the online help to include
information about user-defined functions. It is *not* the main
command for getting information about CoCoA online. For information
about online help in general, enter '?' or '?online help'.
There are two ways to let 'Help' know about a help string associated
with a user-defined function. First, one may use the line 'Help S'
where S is the help string, as the first line of the function definition.
>EXAMPLE
Define AddThree(X)
Help 'adds 3 to its argument';
Return X+3;
EndDefine;
Help('AddThree');
adds 3 to its argument
-------------------------------
F(0);
3
-------------------------------
The second way to provide a help string for 'Help' is to define a
function Help_F where F is the function identifier.
>EXAMPLE<
Define AddFive(X)
Return X+5;
EndDefine;
Define Help_AddFive()
Return 'adds 5 to its argument';
EndDefine;
Help('AddFive');
adds 5 to its argument
-------------------------------
AddFive(0);
5
-------------------------------
See also:
Define
Online Help
Hilbert
SYNTAX
Hilbert(R:RING or TAGGED("Quotient")):TAGGED("$cocoa/hp.Hilbert")
Hilbert(R:RING or TAGGED("Quotient"),N:INT):INT
SUMMARY
the Hilbert function
DESCRIPTION
The first form of this function computes the Hilbert function for R.
The second form computes the N-th value of the Hilbert function. The
weights of the indeterminates of R must all be 1. If the input is not
homogeneous, the Hilbert function of the corresponding leading term
(initial) ideal or module is calculated. For repeated evaluations of
the Hilbert function, use 'EvalHilbertFn' instead of 'Hilbert(R,N)' in
order to speed up execution.
The coefficient ring must be a field.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
Hilbert(R/Ideal(z^2-xy,xz^2+t^3));
H(0) = 1
H(1) = 4
H(t) = 6t-3 for t >= 2
-------------------------------
M := R^2/Module([x^2-t,xy-z^3],[zy,tz-x^3y+3]);
Hilbert(M);
H(0) = 2
H(1) = 8
H(2) = 20
H(3) = 39
H(t) = 3t^2 + 6t-7 for t >= 4
-------------------------------
Hilbert(M,3)
39
-------------------------------
Hilbert(M,5);
98
-------------------------------
See also:
EvalHilbertFn
HilbertPoly
HVector
Poincare
HilbertPoly
SYNTAX
Hilbert(R:RING or TAGGED("Quotient")):POLY in the ring Qt.
SUMMARY
the Hilbert polynomial
DESCRIPTION
This function returns the Hilbert polynomial for R as a polynomial in
the standard CoCoA ring Qt (= Q[t]).
The weights of the indeterminates of R must all be 1, and the
coefficient ring must be a field.
If the input is not homogeneous, the Hilbert polynomial of the
corresponding leading term (initial) ideal or module is calculated.
For the Hilbert *function*, see 'Hilbert'.
>EXAMPLE<
Use R ::= Q[w,x,y,z];
I := Ideal(z^2-xy,xz^2+w^3);
Hilbert(R/I);
H(0) = 1
H(1) = 4
H(t) = 6t-3 for t >= 2
-------------------------------
F := HilbertPoly(R/I);
F; -- a polynomial in the ring Qt
Qt :: 6t-3
-------------------------------
Subst(F,Qt::t,3);
Qt :: 15
-------------------------------
See also:
EvalHilbertFn
Hilbert
HVector
Poincare
HIntersection, HIntersectionList
SYNTAX HIntersection(I_1:IDEAL,...,I_n:IDEAL):IDEAL HIntersectionList(L:LIST of IDEAL):IDEAL SUMMARY intersection of ideals DESCRIPTION The function 'HIntersection' returns the intersection of I_1,...,I_n using a Hilbert-driven algorithm. It differs from 'Intersection' only when the input is inhomogeneous, in which case, 'HIntersection' may be faster. The function 'HIntersectionList' applies the function 'HIntersection' to the elements of a list, i.e., 'HIntersectionList([I_1,...,I_n])' is the same as 'HIntersection(I_1,...,I_n)'. The coefficient ring must be a field. >EXAMPLE< Use R ::= Q[x,y,z]; HIntersection(Ideal(x-z,y-2z),Ideal(x-2z,y-z)); Ideal(x + y - 3z, y^2 - 3yz + 2z^2) ------------------------------- L := [Ideal(x-z,y-2z),Ideal(x-2z,y-z)]; HIntersectionList(L); Ideal(x + y - 3z, y^2 - 3yz + 2z^2) ------------------------------- See also: Intersection, IntersectionList
Homogenized
SYNTAX
Homogenized(X:INDET,E:T):T
where T is of type IDEAL or POLY, or T is a LIST recursively
constructed of types IDEAL, POLY, and LIST.
SUMMARY
homogenize with respect to an indeterminate
DESCRIPTION
This function returns the homogenization of E with respect to the
indeterminate X, which must have weight 1. Note that in the case
where E is an ideal, 'Homogenized' returns the ideal generated by the
homogenizations of all the elements of E, not just the homogenization
of the generators of E (see the example, below). The coefficient ring
must be a field for this function to work reliably.
>EXAMPLE<
Use R ::= Q[x,y,z,w];
Homogenized(w,x^3-y);
x^3 - yw^2
-------------------------------
Homogenized(w,[x^3-y,x^4-z]);
[x^3 - yw^2, x^4 - zw^3]
-------------------------------
I := Ideal(x^3-y,x^4-z);
Homogenized(w,I); -- don't just get the homogenizations of
-- the generators of I
Ideal(x^3 - yw^2, -xy + zw, x^2z - y^2w, y^3 - xz^2)
-------------------------------
Homogenized(w,[[I,y-z^2],z-y^4]);
[[Ideal(x^3 - yw^2, -xy + zw, x^2z - y^2w, y^3 - xz^2), -z^2 + yw], -y^4 + zw^3]
-------------------------------
HVector
SYNTAX
HVector(R:RING or TAGGED("Quotient")):LIST
SUMMARY
the h-vector of a ring or quotient object
DESCRIPTION
This function returns the h-vector of the ring R, i.e., the
coefficients of the numerator of the simplified Poincare series for R.
The weights of the indeterminates of the current ring must all be 1,
and the coefficient ring must be a field.
If the input is not homogeneous, the Hilbert function of the
corresponding leading term (initial) ideal or module is calculated.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
HVector(R/Ideal(x,y,z)^5);
[1, 3, 6, 10, 15]
-------------------------------
Poincare(R/Ideal(x,y,z)^5);
(1 + 3t + 6t^2 + 10t^3 + 15t^4) / (1-t)
-------------------------------
See also:
Hilbert
Poincare
Ideal
SYNTAX Ideal(P_1:POLY,...,P_n:POLY):IDEAL Ideal(L:LIST):IDEAL Ideal(M:MODULE):IDEAL where L is a list of polynomials and M is contained in a free module of rank 1. SUMMARY convert an expression into an ideal DESCRIPTION The first form returns the ideal generated by P_1,...P_n. The second form returns the ideal generated by the polynomials in L. The third form returns the ideal generated by the polynomials in M; it is the same as 'Cast(M,IDEAL), and requires that the module be a submodule of the free module of rank 1'. >EXAMPLE< Use R ::= Q[x,y,z]; I := Ideal(x-y^2,xy-z); I; Ideal(-y^2 + x, xy - z) ------------------------------- L := [xy-z,x-y^2]; J := Ideal(L); I = J; TRUE ------------------------------- M := Module([y^3-z],[x-y^2]); Ideal(M) = I; TRUE -------------------------------
IdealAndSeparatorsOfPoints
SYNTAX
IdealAndSeparatorsOfPoints(Points:LIST):RECORD
where Points is a list of lists of coefficients representing a set of
*distinct* points in affine space.
SUMMARY
ideal & separators for affine points
DESCRIPTION
This function computes the results of 'IdealOfPoints' and
'SeparatorsOfPoints' together at a cost lower than making the two
separate calls. The result is a record with three fields:
Points -- the points given as argument
Ideal -- the result of IdealOfPoints
Separators -- the result of SeparatorsOfPoints
Thus, if the result is stored in a variable with identifier X, then:
X.Points will be the input list of points; X.Ideal will be the ideal
of the set of points, with generators forming the reduced Groebner basis
for the ideal; and X.Separators will be a list of polynomials whose
i-th element will take the value 1 on the i-th point and 0 on the
others.
NOTE:
* the current ring must have at least as many indeterminates as the
dimension of the space in which the points lie;
* the base field for the space in which the points lie is taken to be
the coefficient ring, which should be a field;
* in the polynomials returned, the first coordinate in the space is
taken to correspond to the first indeterminate, the second to the
second, and so on;
* if the number of points is large, say 100 or more, the returned
value can be very large. To avoid possible problems when printing
such values as a single item we recommend printing out the elements
one at a time as in this example:
X:=IdealAndSeparatorsOfPoints(Pts);
Foreach Element In Gens(X.Ideal) Do
PrintLn Element;
EndForeach;
For ideals and separators of points in projective space, see
'IdealAndSeparatorsOfProjectivePoints'.
>EXAMPLE<
Use R ::= Q[x,y];
Points := [[1, 2], [3, 4], [5, 6]];
X := IdealAndSeparatorsOfPoints(Points);
X.Points;
[[1, 2], [3, 4], [5, 6]]
-------------------------------
X.Ideal;
Ideal(x - y + 1, y^3 - 12y^2 + 44y - 48)
-------------------------------
X.Separators;
[1/8y^2 - 5/4y + 3, -1/4y^2 + 2y - 3, 1/8y^2 - 3/4y + 1]
-------------------------------
See also:
GBM, HGBM
GenericPoints
IdealAndSeparatorsOfProjectivePoints
IdealOfPoints
IdealOfProjectivePoints
Interpolate
QuotientBasis
SeparatorsOfPoints
SeparatorsOfProjectivePoints
IdealAndSeparatorsOfProjectivePoints
SYNTAX
IdealAndSeparatorsOfProjectivePoints(Points:LIST):RECORD
where Points is a list of lists of coefficients representing a set of
*distinct* points in projective space.
SUMMARY
ideal & separators for points
DESCRIPTION
This function computes the results of 'IdealOfProjectivePoints' and
'SeparatorsOfProjectivePoints' together at a cost lower than making
the two separate calls. The result is a record with three fields:
Points -- the points given as argument
Ideal -- the result of IdealOfProjectivePoints
Separators -- the result of SeparatorsOfProjectivePoints
Thus, if the result is stored in a variable with identifier X, then:
X.Ideal will be the ideal of the set of points, with generators
forming a reduced Groebner basis for the ideal; and X.Separators will
be a list of homogeneous polynomials whose i-th element will be
non-zero (actually 1, using the given representatives for the
coordinates of the points) on the i-th point and 0 on the others.
NOTE:
* the current ring must have at least one more indeterminate than the
dimension of the projective space in which the points lie, i.e, at
least as many indeterminates as the length of an element of
the input, Points;
* the base field for the space in which the points lie is taken to be
the coefficient ring, which should be a field;
* in the polynomials returned, the first coordinate in the space is
taken to correspond to the first indeterminate, the second to the
second, and so on;
* if the number of points is large, say 100 or more, the returned
value can be very large. To avoid possible problems when printing
such values as a single item we recommend printing out the elements
one at a time as in this example:
X:=IdealAndSeparatorsOfProjectivePoints(Pts);
Foreach Element In Gens(X.Ideal) Do
PrintLn Element;
EndForeach;
For ideals and separators of points in affine space, see
'IdealAndSeparatorsOfPoints'.
>EXAMPLE<
Use R ::= Q[x,y,z];
Points := [[0,0,1],[1/2,1,1],[0,1,0]];
X := IdealAndSeparatorsOfProjectivePoints(Points);
X.Points;
[[0, 0, 1], [1, 1, 1], [0, 1, 0]]
-------------------------------
X.Ideal;
Ideal(xz - 1/2yz, xy - 1/2yz, x^2 - 1/4yz, y^2z - yz^2)
-------------------------------
X.Separators;
[-2x + z, x, -2x + y]
-------------------------------
See also:
GBM, HGBM
GenericPoints
IdealAndSeparatorsOfPoints
IdealOfPoints
IdealOfProjectivePoints
Interpolate
QuotientBasis
SeparatorsOfPoints
SeparatorsOfProjectivePoints
IdealOfPoints
SYNTAX
IdealOfPoints(Points:LIST):IDEAL
where Points is a list of lists of coefficients representing a set of
*distinct* points in affine space.
SUMMARY
ideal of a set of affine points
DESCRIPTION
This function computes the reduced Groebner basis for the ideal of all
polynomials which vanish at the given set of points. It returns the
ideal generated by that Groebner basis.
NOTE:
* the current ring must have at least as many indeterminates as the
dimension of the space in which the points lie;
* the base field for the space in which the points lie is taken to be
the coefficient ring, which should be a field;
* in the polynomials returned, the first coordinate in the space is
taken to correspond to the first indeterminate, the second to the
second, and so on;
* if the number of points is large, say 100 or more, the returned
value can be very large. To avoid possible problems when printing
such values as a single item we recommend printing out the elements
one at a time as in this example:
I:=IdealOfPoints(Pts);
Foreach Element In Gens(I) Do
PrintLn Element;
EndForeach;
For ideals of points in projective space, see
'IdealOfProjectivePoints'.
>EXAMPLE<
Use R ::= Q[x,y];
Points := [[1, 2], [3, 4], [5, 6]];
I := IdealOfPoints(Points);
I;
Ideal(x - y + 1, y^3 - 12y^2 + 44y - 48)
-------------------------------
I.Gens; -- the reduced Groebner basis
[x - y + 1, y^3 - 12y^2 + 44y - 48]
-------------------------------
See also:
GBM, HGBM
GenericPoints
IdealAndSeparatorsOfPoints
IdealAndSeparatorsOfProjectivePoints
IdealOfProjectivePoints
Interpolate
QuotientBasis
SeparatorsOfPoints
SeparatorsOfProjectivePoints
IdealOfProjectivePoints
SYNTAX
IdealOfProjectivePoints(Points:LIST):IDEAL
where Points is a list of lists of coefficients representing a set of
*distinct* points in projective space.
SUMMARY
ideal of a set of projective points
DESCRIPTION
This function computes the reduced Groebner basis for the ideal of
all homogeneous polynomials which vanish at the given set of points.
It returns the ideal generated by that Groebner basis.
NOTE:
* the current ring must have at least one more indeterminate than the
dimension of the projective space in which the points lie, i.e, at
least as many indeterminates as the length of an element of
the input, Points;
* the base field for the space in which the points lie is taken to be
the coefficient ring, which should be a field;
* in the polynomials returned, the first coordinate in the space is
taken to correspond to the first indeterminate, the second to the
second, and so on;
* if the number of points is large, say 100 or more, the returned
value can be very large. To avoid possible problems when printing
such values as a single item we recommend printing out the elements
one at a time as in this example:
I:=IdealOfProjectivePoints(Pts);
Foreach Element In Gens(I) Do
PrintLn Element;
EndForeach;
For ideals of points in affine space, see 'IdealOfPoints'.
>EXAMPLE<
Use R ::= Q[x,y,z];
I := IdealOfProjectivePoints([[0,0,1],[1/2,1,1],[0,1,0]]);
I;
Ideal(xz - 1/2yz, xy - 1/2yz, x^2 - 1/4yz, y^2z - yz^2)
-------------------------------
I.Gens; -- the reduced Groebner basis
[xz - 1/2yz, xy - 1/2yz, x^2 - 1/4yz, y^2z - yz^2]
-------------------------------
See also:
GBM, HGBM
GenericPoints
IdealAndSeparatorsOfPoints
IdealAndSeparatorsOfProjectivePoints
IdealOfPoints
Interpolate
QuotientBasis
SeparatorsOfPoints
SeparatorsOfProjectivePoints
Identity
SYNTAX Identity(N:INT):MAT SUMMARY the identity matrix DESCRIPTION This function returns the NxN identity matrix. >EXAMPLE< Identity(3); Mat[ [1, 0, 0], [0, 1, 0], [0, 0, 1] ] -------------------------------
If
SYNTAX If B Then C EndIf If B_1 Then C_1 Else C_2 EndIf If B_1 Then C_1 Elsif B_2 Then C_2 Elsif ... EndIf If B_1 Then C_1 Elsif B_2 Then C_2 Elsif ... Else C_r EndIf D If B where the B's are boolean expressions, the C's are command sequences, and D is a single command. SUMMARY conditional statement DESCRIPTION If B_n is the first in the sequence of B_i's to evaluate to TRUE, then C_n is executed. If none of the B_i's evaluates to TRUE, nothing is done. The construct, 'Elsif B Then C' can be repeated any number of times. Note: be careful not to type 'Elseif' by mistake (it has an extraneous 'e'). In the last form, the single command D is performed if B evaluates to TRUE. NOTE: the use of this form is discouraged. It will probably disappear from future versions of CoCoA. For a conditional "expression", assignable to a variable, see 'Cond'. >EXAMPLE< Define Sign(A) If A > 0 Then Return 1 Elsif A = 0 Then Return 0 Else Return -1 EndIf End; Sign(3); 1 ------------------------------- See also: Cond
Image
SYNTAX
Image(R::E:OBJECT,F:TAGGED("RMap")):OBJECT
Image(V:OBJECT,F:TAGGED("RMap")):OBJECT
where R is the identifier for a ring and F has the form
RMap(F_1:POLY,...,F_n:POLY) or the form RMap([F_1:POLY,...,F_n:POLY]).
The number n is the number of indeterminates of the ring R. In the
second form, V is a variable containing a CoCoA object dependent on R
or not dependent on any ring.
SUMMARY
ring homomorphism
DESCRIPTION
This function maps the object E from one ring to another as
determined by F. Suppose the current ring is S, and E is an object
dependent on a ring R; then
Image(R::E,F)
returns the object in S obtained by substituting F_i for the i-th
indeterminate of R in E. Effectively, we get the image of E under
the ring homomorphism,
F: R ---> S
x_i |--> F_i,
where x_i denotes the i-th indeterminate of R.
Notes:
1. The coefficient rings for the domain and codomain must be the same.
2. If R = S, one may use 'Image(E,F)' but in this case it may be
easier to use 'Eval' or 'Subst'.
3. The exact domain is never specified by the mapping F. It is only
necessary that the domain have the same number of indeterminates as F
has components. Thus, we are abusing terminology somewhat in
calling F a map.
4. The second form of the function does not require the prefix 'R::'
since the prefix is associated automatically.
5. If the object E in R is a polynomial or rational function (or list,
matrix, or vector of these) that only involves indeterminates that are
already in S, the object E can be mapped over to S without change
using the command 'BringIn'.
>EXAMPLE<
Use C ::= Q[u,v]; -- domain
Use B ::= Q[x,y]; -- another possible domain
I := Ideal(x^2-y); -- an ideal in B
Use A ::= Q[a,b,c]; -- codomain
F := RMap(a,c^2-ab);
Image(B::xy, F); -- the image of xy under F:B --> A
-a^2b + ac^2
-------------------------------
Image(C::uv,F); -- the image of uv under F:C --> A
-a^2b + ac^2
-------------------------------
Image(I,F); -- the image of the ideal I under F: B --> A
Ideal(a^2 + ab - c^2)
-------------------------------
I; -- the prefix 'B::' was not need in the previous example since
-- I is already labeled by B
B :: Ideal(x^2 - y)
-------------------------------
Image(B::Module([x+y,xy^2],[x,y]),F); -- the image of a module
Module([-ab + c^2 + a, a^3b^2 - 2a^2bc^2 + ac^4], [a, -ab + c^2])
-------------------------------
X := C:: u+v; -- X is a variable in the current ring (the codomain), A,
X; -- whose value is an expression in the ring C.
C :: u + v
-------------------------------
Image(X,F); -- map X to get a value in C
-ab + c^2 + a
-------------------------------
See also:
Accessing Other Rings
BringIn
QZP, ZPQ
Ring Mappings: the Image Function
Subst
Using
In
SYNTAX [E:OBJECT | X In L:LIST And B:BOOL]:LIST [X In L:LIST | B:BOOL]:LIST [E:OBJECT | X In L] where X is a variable identifier which may occur in B or E. SUMMARY create a list satisfying given conditions DESCRIPTION In the first form, E is an arbitrary CoCoA expression and B is a boolean expression, both of which are functions of the variable X. Write E(X) for E and B(X) for B. The first listed command then returns the list of all E(X) such that X is in the list L and B(X) evaluates to TRUE. >EXAMPLE< [[X^2,X^3] | X In [-2,-1,0,1,2] And X <> 0]; [[4, -8], [1, -1], [1, 1], [4, 8]] ------------------------------- [X In [1,2] >< [2,3,4] | X[1]+X[2]=4]; [[1, 3], [2, 2]] ------------------------------- (Note: the >< operator is used to form Cartesian products; it is not the same as the 'not equal' operator, <>.) The second form of the command is the same as the first with E = X. >EXAMPLE< [X In [1,2,3] | X > 1]; [2, 3] ------------------------------- The third form is the same as the first with B = TRUE. >EXAMPLE< [X^2 | X In [1,2,3]]; [1, 4, 9] ------------------------------- See also: IsIn NewList
Indet
SYNTAX Indet(N:INT):POLY SUMMARY individual indeterminates DESCRIPTION This function returns the N-th indeterminate of the current ring. >EXAMPLE< Use R ::= Q[x,y,z]; Indet(2); y ------------------------------- See also: IndetInd IndetIndex IndetName Indets NumIndets
IndetInd
SYNTAX IndetInd(X:INDET):LIST SUMMARY the index of an indeterminate DESCRIPTION This function returns the index of the indeterminate X. >EXAMPLE< Use R ::= Q[x[1..3,1..2],y,z]; IndetInd(x[3,2]); [3, 2] ------------------------------- IndetInd(y); [ ] ------------------------------- See also: Indet IndetIndex IndetName Indets NumIndets
IndetIndex
SYNTAX IndetIndex(X:INDET):INT SUMMARY index of an indeterminate DESCRIPTION This function returns the index of the named determinate. The index is determined by the order in which the indeterminate is listed when the corresponding ring was created. >EXAMPLE< Use R ::= Q[x,y,z] IndetIndex(y); 2 ------------------------------- Use R ::= Q[x[1..2,1..2],y[1..2]]; Indets(); [x[1,1], x[1,2], x[2,1], x[2,2], y[1], y[2]] ------------------------------- IndetIndex(x[2,1]); 3 ------------------------------- S ::= Q[a,b,c]; IndetIndex(S::b); 2 ------------------------------- See also: Indet IndetInd IndetName Indets NumIndets
Indets
SYNTAX Indets():LIST SUMMARY list of current indeterminates DESCRIPTION This function returns the list of indeterminates of the current ring. >EXAMPLE< Use R ::= Q[x,y,z]; Indets(); [x, y, z] ------------------------------- See also: Indet IndetInd IndetIndex IndetName NumIndets
IndetName
SYNTAX IndetName(X:INDET):STRING SUMMARY the name of an indeterminate DESCRIPTION This function returns the name of the indeterminate X as a string. >EXAMPLE< Use R ::= Q[x,y,z]; IndetName(Indet(2)); y ------------------------------- Type(It); STRING ------------------------------- See also: Indet IndetInd IndetIndex NumIndets
Insert, Remove
SYNTAX Insert(V:LIST,N:INT,E:OBJECT):NULL Remove(V:LIST,N:INT):NULL where V is a variable containing a list. SUMMARY insert or remove an object in a list DESCRIPTION The first function inserts E into the list L as the N-th component. >EXAMPLE< L := ["a","b","d","e"]; Insert(L,3,"c"); L; ["a", "b", "c", "d", "e"] ------------------------------- The second function removes the N-th component from L. (The function 'WithoutNth' returns the list obtained by removing the N-th component of L without affecting L, itself.) >EXAMPLE< Use R ::= Q[x,y,z]; L := Indets(); L; [x, y, z] ------------------------------- Remove(L,2); L; [x, z] ------------------------------- See also: Append WithoutNth
Interpolate
SYNTAX
Interpolate(Points:LIST,Values:LIST):POLY
where Points is a list of lists of coefficients representing a set of
*distinct* points and Values is a list of the same size containing
numbers from the coefficient ring.
SUMMARY
interpolating polynomial
DESCRIPTION
This function returns a multivariate polynomial which takes given
values at a given set of points.
NOTE:
* the current ring must have at least as many indeterminates as the
dimension of the space in which the points lie;
* the base field for the space in which the points lie is taken to be
the coefficient ring, which should be a field;
* in the polynomials returned, the first coordinate in the space is
taken to correspond to the first indeterminate, the second to the
second, and so on;
* if the number of points is large, say 100 or more, the returned
value can be very large. To avoid possible problems when printing
such values as a single item we recommend printing out the elements
one at a time as in this example:
X:=Interpolate(Pts,Vals);
Foreach Element In X Do
PrintLn Element;
EndForeach;
>EXAMPLE<
Use Q[xy];
Points := [[1/2, 2], [3/4, 4], [5, 6/11], [-1/2, -2]];
Values := [1/2,1/3,1/5,-1/2];
F := Interpolate(Points, Values);
F;
-46849/834000y^2 - 1547/52125x + 13418/52125y + 46849/208500
-------------------------------
[Eval(F,P) | P In Points] = Values; -- check
TRUE
-------------------------------
Interreduce, Interreduced
SYNTAX Interreduce(V:LIST of POLY):NULL Interreduce(V:LIST of VECTOR):NULL Interreduced(L:LIST of POLY):LIST of POLY Interreduced(L:LIST of VECTOR):LIST of VECTOR where V is a variable containing a list. SUMMARY interreduce a list of polynomials or vectors DESCRIPTION These functions reduce each polynomial (resp., vector) using the other polynomials (resp., vectors) as reduction rules. The process terminates when each is in normal form with respect to the others. The function 'Interreduce' takes a variable containing a list and overwrites that variable with the interreduced list. The second returns an interreduced list without affecting its arguments. Note that the definition of normal form depends on the current value of the option FullRed of the panel GROEBNER. If FullRed is FALSE it means that a polynomial (resp., vector) is in normal form when its leading term with respect to the current term ordering cannot be reduced. If FullRed is TRUE it means that a polynomial (resp., vector) is in normal form if and only if each monomial cannot be reduced. >EXAMPLE< UnSet FullRed; -- FullRed = FALSE Use R ::= Q[x,y,z]; Interreduced([x^3-xy^2+yz,xy,z]); [x^3 - xy^2 + yz, xy, z] ------------------------------- Set FullRed; -- FullRed = TRUE (the default value) Interreduced([x^3-xy^2+yz,xy,z]); [xy, z, x^3] ------------------------------- L := [x^3-xy^2+yz,xy,z]; Interreduce(L); L; [xy, z, x^3] ------------------------------- See also: FullRed
Intersection, IntersectionList
SYNTAX Intersection(E_1:IDEAL,...,E_n:IDEAL):IDEAL Intersection(E_1:LIST,....,E_n:LIST):LIST Intersection(E_1:MODULE,....,E_n:MODULE):MODULE IntersectionList(L:LIST):OBJECT SUMMARY intersect lists, ideals, or modules DESCRIPTION The function 'Intersection' returns the intersection of E_1,...,E_n. In the case where the E_i's are lists, it returns the elements common to all of the lists. The function 'IntersectionList' applies the function 'Intersection' to the elements of a list, i.e., 'IntersectionList([X_1,...,X_n])' is the same as 'Intersection(X_1,...,X_n)'. The coefficient ring must be a field. NOTE: In order to compute the intersection of inhomogeneous ideals, it may be faster to use the function 'HIntersection'. To compute the intersection of ideals corresponding to zero-dimensional schemes, see the commands 'GBM' and 'HGBM'. >EXAMPLE< Use R ::= Q[x,y,z]; Points := [[0,0],[1,0],[0,1],[1,1]]; -- a list of points in the plane I := Ideal(x,y); -- the ideal for the first point Foreach P In Points Do I := Intersection(I,Ideal(x-P[1]z,y-P[2]z)); EndForeach; I; -- the ideal for (the projective closure of) Points Ideal(y^2 - yz, x^2 - xz) ------------------------------- Intersection(['a','b','c'],['b','c','d']); ["b", "c"] ------------------------------- IntersectionList([Ideal(x,y),Ideal(y^2,z)]); Ideal(yz, xz, y^2) ------------------------------- It = Intersection(Ideal(x,y),Ideal(y^2,z)); TRUE ------------------------------- See also: GBM, HGBM HIntersection, HIntersectionList
Inverse
SYNTAX Inverse(X:OBJECT):OBJECT SUMMARY multiplicative inverse DESCRIPTION This function computes the multiplicative inverse of its argument. It is included for use when writing Inverse(X) comes more naturally than writing X^(-1), though both notations are functionally equivalent. >EXAMPLE< Inverse(Mat([[1,2], [3,4]])); Mat[ [-2, 1], [3/2, -1/2] ] -------------------------------
IO.SprintTrunc
SYNTAX IO.SprintTrunc(E:OBJECT,N:INT):STRING SUMMARY convert to a string and truncate DESCRIPTION This function works like 'Sprint', turning the value of the expression E into a string, but if the string has length greater than N-1, it is truncated and the string '...' is concatenated. This function is useful in formatting reports of results. >EXAMPLE< Use R ::= Q[xy]; I := Ideal(x,y); IO.SprintTrunc(I,4); Idea... ------------------------------- See also: Format Sprint
IsEven, IsOdd
SYNTAX IsEven(N:INT):BOOL IsOdd(N:INT):BOOL SUMMARY test whether an integer is even or odd DESCRIPTION These functions test whether an integer is even or odd. >EXAMPLE< IsEven(3); FALSE ------------------------------- IsOdd(3); TRUE ------------------------------- See also: IsZero
IsHomog
SYNTAX IsHomog(F:POLY or VECTOR):BOOL IsHomog(L:LIST):BOOL IsHomog(I:IDEAL or MODULE):BOOL SUMMARY test whether given polynomials are homogeneous DESCRIPTION The first form of this function returns TRUE if F is homogeneous. The second form returns TRUE if every element of L is homogeneous. Otherwise, they return FALSE. The third form returns TRUE if the ideal/module can be generated by homogeneous elements, and FALSE if not. Homogeneity is with respect to the first row of the weights matrix. >EXAMPLE< Use R ::= Q[x,y]; IsHomog(x^2-xy); TRUE ------------------------------- IsHomog(x-y^2); FALSE ------------------------------- IsHomog([x^2-xy,x-y^2]); FALSE ------------------------------ Use R ::= Q[x,y],Weights(Mat([[2,3],[1,2]])); IsHomog(x^3y^2+y^4); TRUE ------------------------------- Use R ::= Q[x,y]; IsHomog(Ideal(x^2+y,y)); TRUE ------------------------------- See also: Deg MDeg Weights Modifier
IsIn
SYNTAX
E IsIn F
where E and F are CoCoA objects. For a precise description of
allowable objects, see the full online help entry.
SUMMARY
check if one object is contained in another
DESCRIPTION
The semantics of IsIn is explained in the following table:
----------------------------------------------------------------
| POLY IsIn IDEAL checks for ideal membership. |
| VECTOR IsIn MODULE checks for module membership. |
| OBJECT IsIn LIST checks if the list contains the object. |
| STRING IsIn STRING checks if the first string is a substring |
| of the second one. |
----------------------------------------------------------------
IsIn operator
IsNumber
SYNTAX IsNumber(E:OBJECT):BOOL SUMMARY checks if the argument is a number DESCRIPTION This function returns TRUE if E has type INT, RAT, or ZMOD. Otherwise, it returns FALSE. >EXAMPLE< Use R ::= Q[x,y]; IsNumber(x+y); FALSE ------------------------------- IsNumber(3); TRUE ------------------------------- IsNumber(3%5); TRUE -------------------------------
IsPosTo, IsToPos
SYNTAX IsPosTo(R:RING):BOOL IsToPos(R:RING):BOOL where R is an identifier for a ring. SUMMARY checks the module term-ordering of a ring DESCRIPTION The first function determines whether the ring R has the PosTo module term-ordering. The second function determines whether R has the ToPos module term-ordering. (These are the only possible module term-orderings in CoCoA.) >EXAMPLE< Use R ::= Q[x,y,z]; S ::= Q[x,y],PosTo; IsPosTo(R); FALSE ------------------------------- IsToPos(R); TRUE ------------------------------- IsPosTo(S); TRUE ------------------------------- IsToPos(Var(RingEnv())); -- to check the current ring, R TRUE -------------------------------
IsPrime
SYNTAX
IsPrime(N:INT):BOOL
SUMMARY
prime integer test
DESCRIPTION
This function determines whether a small integer is prime.
The range of permitted values for N is the same the range
of permitted values for the NextPrime function: on most
platforms N < 45000 should work fine, on some platforms
considerably larger values will work.
For N greater than the limit an error is raised.
>EXAMPLE<
IsPrime(32003);
TRUE
-------------------------------
IsPrime(10^100);
ERROR: IsPrime: number too large
CONTEXT: Return(Error("IsPrime: number too large"))
-------------------------------
See also:
NextPrime
Isqrt
SYNTAX Isqrt(N:INT):INT SUMMARY computes the (truncated) square root of an integer DESCRIPTION This function computes the square root of an integer. If the argument is not a perfect square it returns the integer part of the square root. >EXAMPLE< Isqrt(16); 4 ------------------------------- Isqrt(99); 9 ------------------------------- Isqrt(-1); ERROR: Expected non-negative INT CONTEXT: Isqrt(-1) -------------------------------
IsTerm
SYNTAX IsTerm(X:POLY or VECTOR):BOOL SUMMARY checks if the argument is a term DESCRIPTION The function determines whether X is a term. For a polynomial, a "term" is a power-product, i.e., a product of indeterminates. Thus, xy^2z is a term, while 4xy^2z and xy+2z^3 are not. For a vector, a term is a power-product times a standard basis vector, e.g., (0,xy^2z,0). >EXAMPLE< Use R ::= Q[x,y,z]; IsTerm(x+y^2); FALSE ------------------------------- IsTerm(x^3yz^2); TRUE ------------------------------- IsTerm(5x^3yz^2); FALSE ------------------------------- IsTerm(Vector(0,0,xyz)); TRUE ------------------------------- IsTerm(Vector(x^2,y^2)); FALSE ------------------------------- IsTerm(5x^2); FALSE -------------------------------
IsZero
SYNTAX IsZero(X:OBJECT):BOOL SUMMARY test whether an object is zero DESCRIPTION This function tests whether its argument is zero; the argument can be of almost any type for which "zero" makes sense. >EXAMPLE< IsZero(23); FALSE ------------------------------- IsZero(3-3); TRUE ------------------------------- Use R ::= Q[x,y]; IsZero(x^2+3y-1); FALSE ------------------------------- IsZero(Ideal(x^2,xy^3)); FALSE ------------------------------- IsZero(Vector(0,0,0)); TRUE ------------------------------- See also: IsEven IsOdd
Jacobian
SYNTAX Jacobian(L:LIST):MAT where L is a list of polynomials. SUMMARY the Jacobian of a list of polynomials DESCRIPTION This function returns the Jacobian matrix of the polynomials in L with respect to all the indeterminates of the current ring. >EXAMPLE< Use R ::= Q[x,y]; L := [x-y,x^2-y,x^3-y^2]; Jacobian(L); Mat[ [1, -1], [2x, -1], [3x^2, -2y] ] -------------------------------
Last
SYNTAX Last(L:LIST):OBJECT Last(L:LIST,N:INT):OBJECT SUMMARY the last N elements of a list DESCRIPTION In the first form, the function returns the last element of L. In the second form, it returns the list of the last N elements of L. >EXAMPLE< L := [1,2,3,4,5]; Last(L); 5 ------------------------------- Last(L,3); [3, 4, 5] ------------------------------- See also: First Head Tail
Latex
SYNTAX
Latex(X:OBJECT):TAGGED("$cocoa/latex.Latex")
SUMMARY
LaTeX formatting
DESCRIPTION
This function returns its argument in a format suitable for inclusion
in a LaTeX document.
>EXAMPLE<
Use R ::= Q[x,y,z];
F := x^3+2y^2z;
Latex(F);
x^{3} + 2y^{2}z
-------------------------------
M := Mat([[1,2],[3,4]]);
Latex(M);
\left( \begin{array}{ll}
1 & 2 \\
3 & 4 \end{array}\right)
-------------------------------
F := (x+y)/(1-z)^3;
Latex(F);
\frac{ - x - y}{z^{3}-3z^{2} + 3z-1}
-------------------------------
Latex(Ideal(x^2,y+z));
( \ x^{2},
y + z\ )
-------------------------------
See also:
Format
Sprint
LC
SYNTAX LC(F:POLY or VECTOR):C where C is one of INT, RAT, or ZMOD. SUMMARY the leading coefficient of a polynomial or vector DESCRIPTION This function returns the leading coefficient of F, as determined by the term-ordering of the ring to which F belongs. >EXAMPLE< Use R ::= Q[x,y]; LC(x+3x^2-5y^2); 3 ------------------------------- LC(Vector(0,5y+6x^2,y^2)); 6 ------------------------------- See also: Coefficients CoeffOfTerm LT
Len
SYNTAX
Len(E:OBJECT):INT
SUMMARY
the length of an object
DESCRIPTION
This function returns the "length" of an object, as summarized in
the table below:
----------------------------------------------
| type | length |
-----------------------------------------------|
| IDEAL | length of Gens(E) |
| INT | 1 |
| LIST | number of items in the list |
| MAT | number of rows of the matrix |
| MODULE | length of Gens(E) |
| POLY | number of monomials |
| RATFUN | if E=F/G, then Len(E)=Len(F)+Len(G) |
| VECTOR | number of components |
----------------------------------------------
The operator 'Len'
>EXAMPLE<
Use R ::= Q[x,y];
L := ['a',2,3,[4,5]];
Len(L);
4
-------------------------------
Len(L[1]);
1
-------------------------------
Len(L[4]);
2
-------------------------------
Len(x^2 + xy + y^2 + xy^4);
4
-------------------------------
Len((x^2+y^2)/(x+y+y^2));
5
-------------------------------
Len((x+y) - (xy/x));
1
-------------------------------
Len(Ideal(x,x^2));
2
-------------------------------
The function 'Size' returns the amount of memory used by the object.
See also:
Count
Len
Size
LinKer
SYNTAX LinKer(M:MAT):LIST where M is a matrix over Q or Z. SUMMARY find the kernel of a matrix DESCRIPTION This function returns a list whose components are lists representing a Z-basis for the kernel of M. Calling the function twice on the same input will not necessarily produce the same output, though in each case, a basis for the kernel is produced. >EXAMPLE< M := Mat([[1,2,3,4],[5,6,7,8],[9,10,11,12]]); LinKer(M); [[1, -1, -1, 1], [0, 1, -2, 1]] ------------------------------- M*Transposed(Mat(It)); Mat[ [0, 0], [0, 0], [0, 0] ] ------------------------------- See also: LinSol
LinSol
SYNTAX LinSol(M:MAT,L:LIST):LIST where M is an m x n matrix over Z or Q, and L is a list of length m with entries in Z or Q (or a list of such lists). SUMMARY find a solution to a linear system DESCRIPTION This function finds a solution to the inhomogeneous system of equations represented by M and L. Specifically, it returns a list, X, of length n with entries in Z or Q, such that M*Transposed(Mat(X)) = Transposed(Mat([L])), if such X exists; otherwise, it returns the empty list. Once a solution is found, all solutions may be found using the function 'LinKer'. NOTE: 'LinSol' can solve several inhomogeneous systems at once. If L has the form [L_1,...,L_k] where each L_i is a list of length m with entries in Z or Q, then LinSol(M,L) returns a list [X_1,...,X_k] where X_i is a solution to the system of linear equations represented by M and L_i. >EXAMPLE< M := Mat([[3,1,4],[1,5,9],[2,6,5]]); L := [123,456,789]; LinSol(M,L); [199/5, 742/5, -181/5] ------------------------------- M*Transposed(Mat([It])); Mat[ [123], [456], [789] ] ------------------------------- LinSol(M,[L,[1,2,3]]); [[199/5, 742/5, -181/5], [4/15, 7/15, -1/15]] ------------------------------- See also: LinKer
List
SYNTAX List(E:OBJECT):LIST where E has type LIST, MAT, or VECTOR. SUMMARY convert an expression into a list DESCRIPTION This function converts the expression E into a list. It is the same as Cast(E,LIST). >EXAMPLE< Use R ::= Q[x,y]; M:=Jacobian([x^2y^2,x+y]); M; Mat[ [2xy^2, 2x^2y], [1, 1] ] ------------------------------- Head(M); ------------------------------- ERROR: Bad parameters CONTEXT: Head(M) ------------------------------- Head(List(M)); [2xy^2, 2x^2y] ------------------------------- The error occurs because the function 'Head' only accepts lists as arguments. See also: NewList
LM
SYNTAX LM(P:POLY or VECTOR):same type as P SUMMARY the leading monomial of a polynomial or vector DESCRIPTION This function returns the leading monomial of P. The monomial includes the coefficient. To get the leading term of P, (which does not included the coefficient), use 'LT'. >EXAMPLE< Use R ::= Q[x,y]; LM(3x^2y+y); 3x^2y ------------------------------- LM(Vector(2x,y)); Vector(2x, 0) ------------------------------- LT(Vector(2x,y)); Vector(x, 0) ------------------------------- See also: LC LPP LT
Log
SYNTAX Log(F:POLY):LIST SUMMARY the list of exponents of the leading term of a polynomial DESCRIPTION This function returns the list of exponents of the leading term of F. >EXAMPLE< Use R ::= Q[x,y,z]; F := x^3y^2z^5+x^2y+xz^4; Log(F); [3, 2, 5] ------------------------------- See also: LT LogToTerm
LogToTerm
SYNTAX LogToTerm(L:LIST):POLY where L is a list of integers. SUMMARY returns a monomial (power-product) with given exponents DESCRIPTION This function returns the power-product whose list of exponents is L. >EXAMPLE< Use R ::= Q[x,y,z]; LogToTerm([2,3,5]); x^2y^3z^5 ------------------------------- Log(It); [2, 3, 5] ------------------------------- See also: Log
LPos
SYNTAX LPos(V:VECTOR):INT SUMMARY the position of the leading power-product in a vector DESCRIPTION This function returns the position of the leading power-product of V. >EXAMPLE< Use R ::= Q[x,y],ToPos; -- ToPos is the default module term-ordering LT(Vector(x,y^2)); Vector(0, y^2) ------------------------------- LPP(Vector(x,y^2)); y^2 ------------------------------- LPos(Vector(x,y^2)); 2 ------------------------------- Use R ::= Q[x,y],PosTo; LT(Vector(x,y^2)); Vector(x, 0) ------------------------------- LPP(Vector(x,y^2)); x ------------------------------- LPos(Vector(x,y^2)); 1 ------------------------------- See also: LM LPP LT
LPP
SYNTAX
LPP(P:POLY or VECTOR):VECTOR
SUMMARY
the leading power-product of a polynomial or vector
DESCRIPTION
This function returns the leading power-product of P.
>EXAMPLE<
Use R ::= Q[x,y];
LPP(3x^2y+y); -- LPP is the same as LT for polynomials
x^2y
-------------------------------
LPP(Vector(2x,y));
x
-------------------------------
LT(Vector(2x,y)); -- Note the difference between LPP and LT
-- for vectors.
Vector(x, 0)
-------------------------------
See also:
LC
LM
LT
LT
SYNTAX LT(E):same type as E where E has type IDEAL, MODULE, POLY, or VECTOR. SUMMARY the leading term of an object DESCRIPTION If E is a polynomial this function returns the leading term of the polynomial E with respect to the term-ordering of the current ring. For the leading monomial, which includes the coefficient, use 'LM'. >EXAMPLE< Use R ::= Q[x,y,z]; -- the default term-ordering is DegRevLex LT(y^2-xz); y^2 ------------------------------- Use R ::= Q[x,y,z], Lex; LT(y^2-xz); xz ------------------------------- If E is a vector, LT(E) gives the leading term of E with respect to the module term-ordering of the current ring. For the leading monomial, which includes the coefficient, use 'LM'. >EXAMPLE< Use R ::= Q[x,y]; V := Vector(0,x,y^2); LT(V); -- the leading term of V w.r.t. the default term-ordering, ToPos Vector(0, 0, y^2) ------------------------------- Use R ::= Q[x,y], PosTo; V := Vector(0,x,y^2); LT(V); -- the leading term of V w.r.t. PosTo Vector(0, x, 0) ------------------------------- If E is an ideal or module, LT(E) returns the ideal or module generated by the leading terms of all elements of E, sometimes called the 'initial' ideal or module. >EXAMPLE< Use R ::= Q[x,y,z]; I := Ideal(x-y,x-z^2); LT(I); Ideal(x, z^2) ------------------------------- See also: LC LM LPP Module Orderings Orderings
?, Man
SYNTAX
?key:NULL
??key:NULL
Man(T:STRING):NULL
Man(T:STRING,0):NULL
where 'key' is a literal string and 0 is the integer 0.
SUMMARY
search online help system
DESCRIPTION
The '?' function is used to search the online help system for
information matching a keyword. In the forms '?key' and '??key', the
'key' is a literal string. This means that 'key' should not appear
enclosed in quotes. For instance, type '?gbasis' rather than
'?"gbasis"'. The command '?' is case insensitive and does not
notice blank space before or after 'key'. Also, the semicolon usually
required at the end of a line of CoCoA input is optional.
The online help contains a manual and a list of commands. Each
section of the manual and each command has an associated list of
keywords. To explain how the search system works, we will use the
following terminology: say the key S 'matches' a keyword K if S is a
substring of K, not counting capitalization. The key S is an 'exact
match' if it is identical to K, not counting capitalization. The
command '?S' searches for keywords in the online help system matching
the S. If only one match occurs or an exact match occurs, the
corresponding information is displayed. Otherwise, if more than one
match occurs (and no exact match), matching keywords are listed. It
is often advisable to make search words small at first to get as many
matches as possible.
>EXAMPLE<
?po
See:
Commands and Functions for Polynomials
DensePoly
EquiIsoDec
Evaluation of Polynomials
Factoring Polynomials
--> Output suppressed <--
?for poly
============ Commands and Functions for Polynomials =============
The following are commands and functions for polynomials:
--> Output suppressed <--
>END OF EXAMPLE<
Intelligent choice of the string S can save a lot of typing. For
example, there are many sections in the manual whose titles begin:
'Commands and Functions for'; in the example above, we matched the
corresponding section for polynomials by choosing the search string
'for poly'.
The command '??S' displays all keywords in the online help system that
match S (an exact match is not required, and *all* keywords are listed
even if there is an exact match).
>EXAMPLE<
??gbasis
See:
GB.Start_GBasis
GBasis
ReducedGBasis
-------------------------------
>END OF EXAMPLE<
Typing '?gbasis', with a single question mark produces only the manual
entry for 'GBasis'.
The function 'Man' is equivalent to '?' except it requires an actual
string, e.g., 'Man("gbasis")' rather than 'Man(gbasis)'. 'Man' with
the optional second argument set to the number 0 is equivalent to
'??'. Since 'Man' requires more typing than '?', there should never
be a need to use it. (The command '?' was introduced in CoCoA 4.2 and
is intended to replace 'Man'.)
Note: The set of keywords associated with any section of the manual
always includes the title of the manual, so it might help to first
take a look at the table of contents, using 'H.Toc'. (The titles of
Parts, which are numbered by 'H.Toc' do *not* appear as keywords: only
titles of chapters and sections.) Similarly, the set of keywords for
a command always includes the command's name. The complete list of
documented commands can be printed by entering 'H.Commands('')'.
See also:
H.Commands
H.Syntax
MapDown
SYNTAX
MapDown(F:POLY):RAT or ZMOD
SUMMARY
convert a constant polynomial to a number
DESCRIPTION
This function converts a constant polynomial to the equivalent
coefficient. If the argument is not a constant polynomial, an error
is signalled.
>EXAMPLE<
Use Q[x,y,z];
Type((x+1)^2 - x*(x+2)); -- value is seen as a polynomial
POLY
-------------------------------
MapDown((x+1)^2 - x*(x+2)); -- attempt to map down to the coeff ring
1
-------------------------------
Type(It); -- value is now simply a coefficient
RAT
-------------------------------
MapDown((x+1)^2 - x^2); -- 2*x + 1 is not a coefficient
ERROR: Cannot MapDown non-const poly
CONTEXT: Error("Cannot MapDown non-const poly")
-------------------------------
Mat
SYNTAX Mat(E):MAT Mat[E]:MAT where E is either: a 'rectangular' lists of lists, a vector, or a module. SUMMARY convert an expression into a matrix DESCRIPTION This function converts the expression E into a matrix. The first form is equivalent to Cast(E,MAT). >EXAMPLE< Use R ::= Q[x,y]; L := [[1,2],[3,4]]; Mat(L); Mat[ [1, 2], [3, 4] ] ------------------------------- M := Module([x,x^2,y],[x^2,y,0]); Mat(M); Mat[ [x, x^2, y], [x^2, y, 0] ] ------------------------------- Mat([[1,2],[3,4]]); -- note the syntax here Mat[ [1, 2], [3, 4] ] ------------------------------- Mat[[1,2],[3,4]]; -- and here Mat[ [1, 2], [3, 4] ] ------------------------------- M:=Mat([['a','b'],['c',[1,2]]]); -- a slightly more obscure example N:=Mat([['d','e'],['f',[3,4]]]); M+N; Mat[ ["ad", "be"], ["cf", [4, 6]] ] ------------------------------- See also: BlockMatrix NewMat
Max, Min
SYNTAX
Max(E_1:OBJECT,...,E_n:OBJECT):OBJECT
Min(E_1:OBJECT,...,E_n:OBJECT):OBJECT
Max(L:LIST):OBJECT
Min(L:LIST):OBJECT
SUMMARY
a maximum or minimum element of a sequence or list
DESCRIPTION
In the first form, these functions return a maximum and minimum,
respectively, of E_1,...,E_n. In the second form, they return a
maximum and minimum, respectively, of the objects in the list L.
>EXAMPLE<
Max([1,2,3]);
3
-------------------------------
Max(1,2,3);
3
-------------------------------
Min(1,2,3);
1
-------------------------------
Use R ::= Q[x,y,z];
Max(x^3z, x^2y^2); -- x^2y^2 > x^3z in the default ordering, DegRevLex
x^2y^2
-------------------------------
Min(x^3z, x^2y^2);
x^3z
-------------------------------
Use R ::= Q[x,y,z], DegLex;
Max(x^3z, x^2y^2); -- x^3z > x^2y^2 in DegLex
x^3z
-------------------------------
Max(Ideal(x),Ideal(x^2),Ideal(x,y),Ideal(x-2,y-1)); -- ordered by inclusion
-- a maximal element in the list is returned
Ideal(x, y)
-------------------------------
See also:
Relational Operators
MDeg
SYNTAX MDeg(F:POLY):LIST SUMMARY multi-degree of an polynomial DESCRIPTION This function returns the multi-degree of F, as determined by the weights of the current ring. The function 'Deg' returns the weight given by the first row of the weights matrix. >EXAMPLE< Use R ::= Q[x,y], Weights(Mat([[1,2],[3,4],[5,6]])); MDeg(x); [1, 3, 5] ------------------------------- MDeg(y); [2, 4, 6] ------------------------------- Deg(y); 2 ------------------------------- MDeg(x^2+y); [2, 6, 10] ------------------------------- See also: Deg Weights Modifier
Memory
SYNTAX
Memory():TAGGED("Memory")
Memory(R:RING):TAGGED("Memory")
SUMMARY
contents of local memory or ring-bound memory
DESCRIPTION
The first form of this function prints the contents of the working
memory, i.e, all non-global variables. The second form lists all
global variables bound to the ring R.
For more information about memory in CoCoA, see the chapter entitled
"Memory Management."
>EXAMPLE<
Use R ::= Q[x,y,z];
I := Ideal(x-y^2,xy-z^3);
X := 14;
ENV.R.Y := 5; -- a global variable bound to R
-- recall that 'ENV.R' is equivalent to 'MEMORY.ENV.R'
Use S ::= Q[a,b];
J := Ideal(a,b);
ENV.S.Z := 7;
Memory();
["I", "J", "X"]
-------------------------------
Memory(R);
["Y"]
-------------------------------
See also:
GlobalMemory
Memory Management
MinGens
SYNTAX
MinGens(M:IDEAL or MODULE or TAGGED("Quotient")):LIST
SUMMARY
list minimal generators
DESCRIPTION
If M is an ideal or module, this function returns a list of minimal
generators for M. If M is the quotient of the current ring by an
ideal I or the quotient of a free module by the submodule N, then
MinGens returns a set of minimal generators for I or N, respectively.
The coefficient ring must be a field.
The input must be homogeneous. The similar command 'Minimalized',
will accept inhomogeneous input.
>EXAMPLE<
Use R ::= Q[x,y,z];
I:=Ideal(x-y,(x-y)^4,z+y,(z+y)^2);
I;
Ideal(x - y, x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4, y + z, y^2 + 2yz + z^2)
-------------------------------
MinGens(I);
[y + z, x + z]
-------------------------------
MinGens(R/I);
[y + z, x + z]
-------------------------------
M :=Module([x+y,x-y],[(x+y)^2,(x+y)(x-y)]);
MinGens(M);
[Vector(x + y, x - y)]
-------------------------------
MinGens(R^2/M);
[Vector(x + y, x - y)]
-------------------------------
See also:
Minimalize, Minimalized
Minimalize, Minimalized
SYNTAX Minimalize(X:IDEAL):NULL Minimalize(X:MODULE):NULL Minimalized(E:IDEAL):IDEAL Minimalized(E:MODULE):MODULE where X is a variable containing an ideal or module. SUMMARY remove redundant generators DESCRIPTION In the inhomogeneous case the first form removes redundant generators from the ideal or module contained in X, storing the result in X; the original ideal or module is overwritten. In the inhomogeneous case the second form returns the ideal or module obtained by removing redundant generators from E. In the homogeneous case, one obtains a generating set with smallest possible cardinality. The minimal set of generators found by CoCoA is not necessarily a subset of the given generators. As with the inhomogeneous case, the first form overwrites the ideal or module contained in X and the second returns the minimalized ideal or module. The coefficient ring is assumed to be a field. >EXAMPLE< Use R ::= Q[x,y,z]; I := Ideal(x-y^2,z-y^5,x^5-z^2); I; Ideal(-y^2 + x, -y^5 + z, x^5 - z^2) ------------------------------- Minimalized(I); Ideal(-y^2 + x, -y^5 + z) ------------------------------- I; Ideal(-y^2 + x, -y^5 + z, x^5 - z^2) ------------------------------- Minimalize(I); I; Ideal(-y^2 + x, -y^5 + z) ------------------------------- J := Ideal(x, x-y, y-z, z^2); Minimalized(J); Ideal(y - z, x - z, z) ------------------------------- See also: MinGens
Minors
SYNTAX Minors(N:INT,M:MAT):LIST SUMMARY list of minor determinants of a matrix DESCRIPTION This function returns the list of all determinants of N x N submatrices of M. >EXAMPLE< M := Mat([[1,2,3],[-1,2,4]]); Minors(2,M); [4, 7, 2] ------------------------------- See also: Det
MinSyzMinGens
SYNTAX MinSyzMinGens: FUNCTION ELIMINATED. SUMMARY minimal generators of syzygies of minimal generators DESCRIPTION The MinSyzMinGens function has been removed. See also: SyzMinGens
Mod2Rat
SYNTAX Mod2Rat(Residue:INT, Modulus:INT, DenomBound:INT):RAT SUMMARY reconstructing rationals from modular integers DESCRIPTION This function determines a rational number equivalent to the given residue modulo the given modulus; the denominator will not exceed DenomBound, and the absolute value of the numerator will be at most Modulus/(2*DenomBound) -- this guarantees uniqueness of the answer. If no such rational exists, or if DenomBound exceeds Modulus/2 then zero is returned. DenomBound must be positive; if not, zero is returned. >EXAMPLE< Mod2Rat(239094665,314159265,10000); 355/113 ------------------------------- Residue := 1234567; -- To compute inverse of Modulus := 2^100; -- Residue modulo Modulus, Mod2Rat(Residue, Modulus, Div(Modulus-1,2));-- result is the denominator. 1/284507170216309247716413542199 ------------------------------- See also: Mod
Module
SYNTAX Module(V_1:VECTOR,...,V_n:VECTOR):MODULE Module(L:LIST):MODULE Module(I:IDEAL):MODULE where L is a list of elements cast-able to vectors. SUMMARY convert an expression into a module DESCRIPTION The first function returns the module generated by the vectors V_1,...,V_n. The second function returns the module generated by the (elements cast-able to) vectors in L. The third function returns the module generated by the generators of I. This is the same as the function 'Cast(E,MODULE)'. >EXAMPLE< Use R ::= Q[x,y,z]; M := Module([x^2-y^3,x^3-yz],[x^3,z^2]); M; Module([-y^3 + x^2, x^3 - yz], [x^3, z^2]) ------------------------------- L := [[-y^3 + x^2, x^3 - yz], [x^3, z^2]]; Module(L) = M; TRUE ------------------------------- I := Ideal(x,y^2,z^3); Module(I); Module([x], [y^2], [z^3]) ------------------------------- See also: Cast
Monic
SYNTAX Monic(F:POLY):POLY Monic(L:LIST of POLY):LIST of POLY SUMMARY divide polynomials by their leading coefficients DESCRIPTION In the first form, this function returns F divided by its leading coefficient (see 'LC') or, if F is zero, it returns zero. In its second form, it returns the list obtained by applying the first form of Monic to each of the components of L. >EXAMPLE< Use R ::= Q[x,y]; L := [4x^5-y^2,3x-2y^4]; Monic(L); [x^5 - 1/4y^2, y^4 - 3/2x] ------------------------------- Use R ::= Z[x,y]; -- WARNING: Coeffs are not in a field -- GBasis-related computations could fail to terminate or be wrong ------------------------------- ------------------------------- L := [4x^5-y^2,3x-2y^4]; Monic(L); -- can't invert coefficients over Z ------------------------------- ERROR: Cannot divide CONTEXT: Cond(Type(X) = LIST, [Monic(A)|A IN X], IsZero(X), X, X / LC(X)) ------------------------------- Use R ::= Z/(5)[x,y]; F := 2x^2+4y^3; Monic(F); y^3 - 2x^2 ------------------------------- See also: LC
Monomials
SYNTAX Monomials(F:POLY or VECTOR):LIST SUMMARY the list of monomials of a polynomial or vector DESCRIPTION This function returns the list of monomials of F. The function 'Support' returns the list of terms (monomials without coefficients). >EXAMPLE< Use R ::= Q[xy]; F := 3x^2y+5y^3-xy^5; Monomials(F); [-xy^5, 3x^2y, 5y^3] ------------------------------- Support(F); [xy^5, x^2y, y^3] ------------------------------- Monomials(Vector(3x^2y+y,5xy+4)); [Vector(3x^2y, 0), Vector(0, 5xy), Vector(y, 0), Vector(0, 4)] ------------------------------- See also: Coefficients Support
MonsInIdeal
SYNTAX
MonsInIdeal(I:IDEAL):IDEAL
SUMMARY
ideal generated by the monomials in an ideal
DESCRIPTION
This function returns the ideal generated by all monomials in the
original ideal I.
>EXAMPLE<
Use R ::= Q[x,y,z];
I:=Ideal(xy^3+z^2, y^5-z^3, xz-y^2-x^3, x^4-xz^2+y^3);
MonsInIdeal(I);
Ideal(z^3, yz^2, x^2z^2, x^5z, x^4yz, x^5y, x^2y^2z, x^7, x^4y^2,
xy^3z, y^4z, xy^4, x^3y^3, y^5)
-------------------------------
More
SYNTAX More(S:STRING):NULL More():NULL SUMMARY print a string, N lines at a time DESCRIPTION The purpose of this function is to print the string S without scrolling off of the screen. The first form of this function stores the string S in a 'MoreDevice', then prints the first N lines from the MoreDevice where N is the integer stored in the global variable MEMORY.MoreCount. Subsequent calls to 'More' print the next N lines from the MoreDevice, (each time removing the lines from the device) until the MoreDevice is empty. After each call to 'More' a line with 'More();' is printed as long as the MoreDevice is not empty. This line is easily cut-and-pasted. The user may set the number of lines to print with the command 'MEMORY.MoreCount := X', where X is an integer. The default is value is 20. If E is *any* CoCoA object, the command 'Sprint(E)' converts E into a string. The output may then be given to 'More' for printing. To use the more device with the online help system, see 'H.SetMore, H.UnSetMore'. See also: H.SetMore, H.UnSetMore Commands and Functions for Strings
Multiplicity
SYNTAX
Multiplicity(R:RING or TAGGED("Quotient")):INT
SUMMARY
the multiplicity (degree) of a ring or quotient object
DESCRIPTION
This function computes the multiplicity (or degree) of the ring R,
i.e., the leading coefficient of the Hilbert polynomial multiplied by
the factorial of the degree of the Hilbert polynomial. The weights of
the indeterminates of the current ring must all be 1.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
Multiplicity(R/Ideal(x,y,z)^5);
35
-------------------------------
See also:
Hilbert
Poincare
NewId
SYNTAX NewId():STRING SUMMARY create a new identifier DESCRIPTION This function returns a string of the form 'V#N' where N is an integer. Each time it is called, the integer N changes, producing a new string. The purpose is to produce identifiers for variables or rings. (CoCoA does not check for the unlikely event that variables of the same form have been defined without the use of 'NewId'.) The function 'NewId' is often used with 'Var'. The most important use for this function is for creating temporary rings within user-defined functions. For an example, see the section of the tutorial entitled "Rings Inside User-Defined Functions." >EXAMPLE< NewId(); V#0 ------------------------------- NewId(); V#1 ------------------------------- X := NewId(); X; V#2 ------------------------------- Var(X) := 3; Var(NewId()) := 4; Describe Memory(); ------------[Memory]----------- It = V#2 V#2 = 3 V#3 = 4 X = V#2 ------------------------------- Y := NewId(); Var(Y) ::= Q[a,b]; Use Var(Y); RingEnvs(); ["Q", "Qt", "R", "V#6", "Z"] ------------------------------- Y; V#6 ------------------------------- Var(Y); Q[a,b] ------------------------------- See also: Rings Inside User-Defined Functions Var
NewList
SYNTAX NewList(N:INT):LIST NewList(N:INT,E:OBJECT) SUMMARY create a new list DESCRIPTION The second function returns a list of length N, filled by E. The first function, in which E is not indicated, returns a list of length N filled with 'Null' values. >EXAMPLE< NewList(4,"a"); ["a", "a", "a", "a"] ------------------------------- NewList(4); [Null, Null, Null, Null] ------------------------------- See also: List
NewMat
SYNTAX NewMat(M:INT,N:INT):MAT NewMat(M:INT,N:INT,E:OBJECT):MAT SUMMARY create a new matrix DESCRIPTION The second function returns an MxN matrix, filled by E. The first function, in which E is not indicated, returns an MxN matrix filled with 'Null' values. >EXAMPLE< NewMat(2,3,"a"); Mat[ ["a", "a", "a"], ["a", "a", "a"] ] ------------------------------- NewMat(2,2); Mat[ [Null, Null], [Null, Null] ] ------------------------------- See also: Mat
NewVector
SYNTAX NewVector(N:INT):VECTOR NewVector(N:INT,E:OBJECT):VECTOR where E is cast-able to POLY. SUMMARY create a new vector DESCRIPTION The second function returns a vector of length N, each of whose components is E. The first function, in which E is not indicated, returns a vector of length N filled with zeros. >EXAMPLE< Use R ::= Q[x,y]; NewVector(4); Vector(0, 0, 0, 0) ------------------------------- NewVector(3,x^2+y); Vector(x^2 + y, x^2 + y, x^2 + y) ------------------------------- See also: Vector
NextPrime
SYNTAX NextPrime(N:INT):INT SUMMARY find the next largest prime number DESCRIPTION This function computes the smallest prime number greater than N. If N is negative or too large then the value zero is returned. This function may generate primes larger than permitted as the characteristic of a finite field in CoCoA. On most platforms primes up to about 45000 can be generated; in some cases a higher limit exists. >EXAMPLE< NextPrime(1000); 1009 ------------------------------- See also: IsPrime
NF
SYNTAX NF(F:POLY,I:IDEAL):POLY NF(V:VECTOR,M:MODULE):VECTOR SUMMARY normal form DESCRIPTION The first function returns the normal form of F with respect to I. It also computes a Groebner basis of I if that basis has not been computed previously. The second function returns the normal form of V with respect to M. It also computes a Groebner basis of M if that basis has not been computed previously. The coefficient ring is assumed to be a field. Note that the definition of normal form depends on the current value of the option FullRed of the panel GROEBNER. If FullRed is FALSE it means that a polynomial is in normal form when its leading term with respect to the the current term ordering cannot be reduced. If FullRed is TRUE it means that a polynomial is in NF if and only if each monomial cannot be reduced. >EXAMPLE< Use R ::= Q[x,y,z]; Set FullRed; I := Ideal(z); NF(x^2+xy+xz+y^2+yz+z^2,I); x^2 + xy + y^2 ------------------------------- UnSet FullRed; NF(x^2+xy+xz+y^2+yz+z^2,I); x^2 + xy + y^2 + xz + yz + z^2 ------------------------------- See also: DivAlg FullRed GenRepr IsIn NFsAreZero NR
NFsAreZero
SYNTAX NFsAreZero(L:LIST of POLY,M IDEAL):BOOL NFsAreZero(L:LIST of VECTOR,M:MODULE):BOOL SUMMARY test if normal forms are zero DESCRIPTION This function returns TRUE if each component of L has normal form 0 with respect to M, i.e., if each component is an element of M. Otherwise, it returns FALSE. The coefficient ring is assumed to be a field. >EXAMPLE< Use S ::= Q[t,x,y,z]; I := Ideal(t^31-t^6-x, t^8-y, t^10-z); F := y^5-z^4; G := (t^8-y)(3F+t^10-z); NFsAreZero([F,G],I); -- F and G are in I TRUE ------------------------------- NFsAreZero([F,x,G],I); -- x is not in I FALSE ------------------------------- See also: IsIn NF
NonZero
SYNTAX NonZero(L:LIST or VECTOR):LIST SUMMARY remove zeroes from a list DESCRIPTION This function returns the list obtained by removing the zeroes from L. >EXAMPLE< Use R ::= Q[x,y,z]; NonZero(['a',0,0,3,Ideal(y),0]); ["a", 3, Ideal(y)] ------------------------------- See also: FirstNonZero, FirstNonZeroPos
Not, And, Or
SYNTAX Not E E And F E Or F where E and F are of type BOOL. SUMMARY boolean operators DESCRIPTION These operators have their usual meanings. Note that when two or more boolean expressions are combined with AND, they are evaluated one by one until a FALSE expression is found. The rest are not evaluated. For example, given the expression "A And B", the system does not attempt to evaluate B unless A evaluates to TRUE. Similarly, evaluation of a sequence of boolean expressions connected by OR stops as soon as a TRUE expression is found.
NR
SYNTAX NR(X:POLY,L:LIST of POLY):POLY NR(X:VECTOR,L:LIST of VECTOR):VECTOR SUMMARY normal reduction DESCRIPTION This function returns the normal remainder of X with respect to L, i.e., it returns the remainder from the division algorithm. To get both the quotients and the remainder, use 'DivAlg'. Note that if the list does not form a Groebner basis, the remainder may not be zero even if X is in the ideal or module generated by L (use 'GenRepr' or 'NF' instead). >EXAMPLE< Use R::= Q[xyz]; F := x^2y+xy^2+y^2; NR(F,[xy-1,y^2-1]); x + y + 1 ------------------------------- V := Vector(x^2+y^2+z^2,xyz); NR(V,[Vector(x,y),Vector(y,z),Vector(z,x)]); Vector(z^2, z^3 - yz - z^2) ------------------------------- See also: DivAlg GenRepr NF
Num, Den
SYNTAX Num(N:INT or RAT):INT Den(N:INT or RAT):INT Num(N:POLY or RATFUN):POLY Den(N:POLY or RATFUN):POLY SUMMARY numerator, denominator DESCRIPTION These functions return the numerator and denominator of N. The numerator and denominator can also be found using '.Num' and '.Den'. >EXAMPLE< Num(3/2); 3 ------------------------------- Den(x/(x+y)); x + y ------------------------------- X := 2/3; X.Num; X.Den; 2 ------------------------------- 3 ------------------------------- See also: Numerators and Denominators for Rational Functions Numerators and Denominators for Rational Numbers
NumComps
SYNTAX NumComps(X:VECTOR or MODULE):INT SUMMARY the number of components of a vector DESCRIPTION If X is a vector, this function returns the number of components of X; it gives the same result as Len(X). If X is a module, then this function returns the rank of the free module in which X is defined. >EXAMPLE< Use R ::= Q[x,y]; NumComps(Vector(x,y,x^2+y^2,x^2-y^2)); 4 ------------------------------- M := Module([x,y^2,2+x^2y],[x,0,y]); -- a submodule of R^3 NumComps(M); 3 ------------------------------- M.NumComps; -- alternative syntax 3 ------------------------------- See also: Len
NumIndets
SYNTAX NumIndets():INT NumIndet(R:RING):INT SUMMARY number of indeterminates DESCRIPTION This function returns the number of indeterminates of the current ring or of R. >EXAMPLE< S ::= Q[x,y]; Use R ::= Q[x,y,z]; NumIndets(); 3 ------------------------------- NumIndets(S); 2 ------------------------------- See also: Indet IndetInd IndetIndex IndetName Indets
OpenIFile, OpenOFile
SYNTAX
OpenIFile(S:STRING):DEVICE
OpenOFile(S:STRING):DEVICE
OpenOFile(S:STRING,'w' or 'W'):DEVICE
SUMMARY
open input or output file
DESCRIPTION
These functions open files for input or output. 'OpenIFile' opens
the file with name S. Input from that file can then be read with
'Get'. 'OpenOFile' opens the file with name S---creating it if it
does not already exist---for output. The function 'Print On' is then
used for writing output to the file. If OpenOFile is used without a
second argument or if the second argument is not 'w' or 'W' then
'Print On' will append output to the file. Otherwise, any existing
file with the name S will be erased before the output is written.
(Note: one would normally use 'Source' to read CoCoA commands from a file.)
>EXAMPLE<
D := OpenOFile('my-test'); -- open 'my-test' for output from CoCoA
Print 'hello world' On D; -- print string into 'mytest'
Print ' test' On D; -- append to the file 'mytest'
Close(D); -- close the file
D := OpenIFile('my-test'); -- open 'my-test' for input to CoCoA
Get(D,3); -- get the first three characters (in Ascii code)
[104, 101, 108]
-------------------------------
Ascii(It); -- convert the ascii code into characters
hel
-------------------------------
Close(D);
D := OpenOFile('my-test','w'); -- clear 'my-test'
Print 'goodbye' On D; -- 'mytest' now consists only of the string 'goodbye'
Close(D);
See also:
Close
Introduction to IO
OpenIString, OpenOString
Source
OpenIString, OpenOString
SYNTAX
OpenIString(S:STRING,T:STRING):DEVICE
OpenOString(S:STRING):DEVICE
SUMMARY
open input or output string
DESCRIPTION
These functions open strings for input or output. The string S
serves as the name of the device opened for input or output; one may
use the empty string. 'OpenIString' is used to read input from the
string T with the help of 'Get'. 'OpenOString' is used to write to a
string with the help of 'Print On'.
>EXAMPLE<
S := 'hello world';
D := OpenIString('',S); -- open the string S for input to CoCoA
L:= Get(D,7); -- read 7 characters from the string
L; -- ascii code
[104, 101, 108, 108, 111, 32, 119]
-------------------------------
Ascii(L); -- convert ascii code to characters
hello w
-------------------------------
Close(D); -- close device D
D := OpenOString(''); -- open a string for output from CoCoA
L := [1,2,3]; -- a list
Print L On D; -- print to D
D;
Record[Name = "", Type = "OString", Protocol = "CoCoAL"]
-------------------------------
S := Cast(D,STRING); -- S is the string output to D
S; -- a string
[1, 2, 3]
Print ' more characters' On D; -- append to the existing output string
Cast(D,STRING);
[1, 2, 3] more characters
-------------------------------
See also:
Close
Introduction to IO
OpenIFile, OpenOFile
Source
Sprint
OpenLog, CloseLog
SYNTAX
OpenLog(D:DEVICE):NULL
CloseLog(D:DEVICE):NULL
SUMMARY
open or close a log of a CoCoA session
DESCRIPTION
The first function opens the output device D and starts to record the
output from a CoCoA session on D. The second function closes the
device D and stops recording the CoCoA session on D.
At present the choices for the device D are an output file (see
'OpenOFile') or an output string (see 'OpenOString'). Several output
devices may be open at a time. If the panel option 'Echo' is set to
TRUE, both the input and output of the CoCoA session are logged;
otherwise, just the output is logged.
>EXAMPLE<
D := OpenOFile('MySession');
OpenLog(D);
1+1;
2
-------------------------------
G := 1;
Set Echo;
2+2;
2 + 2
4
-------------------------------
F := 2;
F := 2
CloseLog(D);
CloseLog(D)
UnSet Echo;
SET(Echo, FALSE)
The contents of "MySession":
2
-------------------------------
2 + 2
4
-------------------------------
F := 2
CloseLog(D)
See also:
Introduction to IO
OpenIFile, OpenOFile
OpenIString, OpenOString
Set, Unset
Option
SYNTAX Option(O):BOOL where O is a panel option. SUMMARY status of a panel option DESCRIPTION This function returns the status of a panel option (TRUE/FALSE). For a list of panels, use 'Panels' and for the current status of each option for a panel with name P, use 'Panel(P)'. To toggle option values, use 'Set' and 'Unset'. The function 'Option' is particularly useful inside a user-defined function in order to temporarily change the value of an option (restoring the option's original value when the function is complete). See 'Setting Options' for an example of this use of 'Option'. >EXAMPLE< Option(Indentation); FALSE ------------------------------- See also: Introduction to Panels Panel Panels Setting Options Set, Unset
Ord
SYNTAX
Ord():MAT
Ord(R:RING):MAT
Ord(M:MAT):MAT
SUMMARY
matrix defining a term-ordering
DESCRIPTION
The first two forms return matrices which describe the term-ordering
of the current ring or of the ring R, respectively. The last form is
used as a modifier when creating a new ring. In that case, it
determines the term-ordering for the ring (see 'Orderings'). Its
argument is a matrix of integers which defines a term-ordering;
i.e. for a ring with N indeterminates it must be an NxN matrix of full
rank where the first non-zero entry in each column is positive.
>EXAMPLE<
Use S ::= Q[x,y,z], Ord(Mat([[1,0,0],
[0,1,0],
[0,0,1]]));
M := Mat([[1,1],[0,-1]]);
T ::= Q[a,b], Ord(M);
U ::= Z/(101)[x,y,z,t], DegRevLex;
-- The term-order for the current ring, S.
Ord();
Mat[
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]
]
-------------------------------
Ord(T);
Mat[
[1, 1],
[0, -1]
]
-------------------------------
Ord(U);
Mat[
[1, 1, 1, 1],
[0, 0, 0, -1],
[0, 0, -1, 0],
[0, -1, 0, 0]
]
-------------------------------
See also:
DegLex
DegLexMat, DegRevLexMat, LexMat, XelMat
DegRevLex
Elim
Lex
Orderings
Predefined Term-Orderings
Xel
Packages
SYNTAX Packages():LIST SUMMARY list of loaded packages DESCRIPTION This function returns the names of the loaded packages as a list of strings. The string '$cocoa/user' refers to the user-defined functions defined in the current CoCoA session. >EXAMPLE< Packages(); ["$cocoa/builtin", "$cocoa/coclib", "$cocoa/user", "$cocoa/help", "$cocoa/io", "$cocoa/misc"] ------------------------------- See also: CoCoA Packages Supported Packages
Panel
SYNTAX Panel(E) where E is one of 'GENERAL' or 'GROEBNER'. SUMMARY print status of a panel's options DESCRIPTION This command prints the status of the options in one of the three panels. It returns no value. >EXAMPLE< Panel(GENERAL); Echo............... : FALSE Timer.............. : FALSE Trace.............. : FALSE Indentation........ : FALSE TraceSources....... : FALSE SuppressWarnings... : FALSE ComputationStack... : FALSE ------------------------------- See also: Introduction to Panels Option Panels Set, Unset
Panels
SYNTAX Panels() SUMMARY list of CoCoA panels DESCRIPTION This function returns a list of CoCoA's panels. >EXAMPLE< Panels(); ["GENERAL", "GROEBNER"] ------------------------------- Panel(It[1]); Echo............... : FALSE Timer.............. : FALSE Trace.............. : FALSE Indentation........ : FALSE TraceSources....... : FALSE SuppressWarnings... : FALSE ComputationStack... : FALSE ------------------------------- See also: Introduction to Panels Option Panel Set, Unset
Pfaffian
SYNTAX Pfaffian(M:MAT) where M is skew-symmetric. The resulting type depends on the entries of the matrix. SUMMARY the Pfaffian of a skew-symmetric matrix DESCRIPTION This function returns the Pfaffian of M. >EXAMPLE< Use R ::= Q[x,y]; Pfaffian(Mat([[0,y],[-y,0]])); y ------------------------------- See also: Det
PkgName
SYNTAX PkgName():STRING S.PkgName():STRING where S is the identifier or alias for a package. SUMMARY returns the name of a package DESCRIPTION This function returns the (long) name of a package. The first form returns '$cocoa/coclib' and the second returns the name of the package whose name or alias is S. This function is useful as a shorthand, when S is an alias, for the full name a package. >EXAMPLE< GB.PkgName(); $cocoa/gb ------------------------------- $cocoa/gb.PkgName(); $cocoa/gb ------------------------------- PkgName(); $cocoa/coclib -------------------------------
Poincare, HilbertSeries
SYNTAX
Poincare(R:RING or TAGGED("Quotient")):TAGGED("$cocoa/hp.PSeries")
HilbertSeries(R:RING or TAGGED("Quotient")):TAGGED("$cocoa/hp.PSeries")
PoincareShifts(M: Module, ShiftsList: LIST):TAGGED("$cocoa/hp.PSeries")
PoincareShifts(M: TAGGED("Quotient"), ShiftsList: LIST)
:TAGGED("$cocoa/hp.PSeries")
SUMMARY
the Poincare series
DESCRIPTION
These functions all compute the Hilbert-Poincare series of M. The first
two functions perform the same operations. The input, M, must be
homogeneous (with respect to the first row of the weights matrix). In
the standard case, where the weights of all indeterminates are 1, the
result is simplified so that the power appearing in the denominator is
the dimension of M.
NOTES:
(i) the coefficient ring must be a field.
(ii) these functions produce tagged objects: they cannot safely be
(non-)equality to other values.
For more information, see the article: A.M. Bigatti, "Computations of
Hilbert-Poincare Series," J. Pure Appl. Algebra, 119/3 (1997),
237--253.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
Poincare(R);
(1) / (1-t)^4
-------------------------------
Q := R/Ideal(t^2,x,y^3); Poincare(Q);
(1 + 2t + 2t^2 + t^3) / (1-t)
-------------------------------
Poincare(R^2/Module([x^2,y],[z,y]));
(2 + t) / (1-t)^3
-------------------------------
Use R ::= Q[t,x,y,z], Weights([1,2,3,4]);
Poincare(R/Ideal(t^2,x,y^3));
--- Non Simplified Pseries ---
(1-2t^2 + t^4 - t^9 + 2t^11 - t^13) / ( (1-t) (1-t^2) (1-t^3) (1-t^4) )
-------------------------------
Use R ::= Q[t,x,y,z], Weights(Mat([[1,2,3,4],[0,0,5,8]]));
Poincare(R/Ideal(t^2,x,y^3));
--- Non Simplified Pseries ---
( - t^13x^15 + 2t^11x^15 - t^9x^15 + t^4-2t^2 + 1) / ( (1-t) (1-t^2) (1-t^3x^5) (1-t^4x^8) )
-------------------------------
Use P ::= Q[x,y,z];
M := Module([x,y^3], [x-z,0]);
PoincareShifts(M, [2,0]); -- Poincare series of a shifted module
(2x^3) / (1-x)^3
-------------------------------
PoincareShifts(P^2/M, [3,1]); -- Poincare series of a shifted quotient module
(x + x^2 + 2x^3) / (1-x)^2
-------------------------------
See also:
Dim
Hilbert
HVector
Multiplicity
Weights Modifier
WeightsMatrix
Poly
SYNTAX Poly(E:OBJECT):POLY SUMMARY convert an expression into a polynomial DESCRIPTION This function converts the expression E into a polynomial, if possible. It is the same as Cast(E,POLY). >EXAMPLE< F := 3; G := Poly(3); Type(F); INT ------------------------------- Type(G); POLY ------------------------------- Use R ::= Q[x]; Poly(Vector(x)); x ------------------------------- See also: Cast
Print On
SYNTAX
Print E:OBJECT On D:DEVICE
SUMMARY
print to an output device
DESCRIPTION
This command prints the value of expression E to the device D.
Currently, the command can be used to print to files, strings, or the
CoCoA window. In the first two cases, the appropriate device must be
opened with 'OpenOFile' or 'OpenOString'.
>EXAMPLE<
D := OpenOFile('my-test'); -- open 'my-test' for output from CoCoA
Print 'hello world' On D; -- print string into 'mytest'
Close(D); -- close the file
D := OpenIFile('my-test'); -- open 'my-test' for input to CoCoA
Get(D,3); -- get the first three characters (in Ascii code)
[104, 101, 108]
-------------------------------
Ascii(It); -- convert the ascii code into characters
hel
-------------------------------
Close(D);
See 'OpenOFile' for an example using output strings. For printing to
the CoCoA window, just use 'Print E' which is short for 'Print E On
DEV.OUT'.
See also:
Introduction to IO
OpenIFile, OpenOFile
OpenIString, OpenOString
Print, PrintLn
Print, PrintLn
SYNTAX Print E_1,...,E_n :NULL Print(E_1,...,E_n):NULL PrintLn E_1,...,E_n :NULL PrintLn(E_1,...,E_n):NULL where the E_i are CoCoA expressions. SUMMARY print the value of an expression DESCRIPTION The command 'Print' displays the value of each of the expressions, E_i. The parentheses are optional. The argument, 'NewLine', (without quotes, but note the two capital letters), moves the cursor to the next line. The command 'PrintLn' is equivalent to 'Print' with a final extra argument, 'NewLine'; in other words, it prints the values of its arguments, then moves the cursor to the next line. The parentheses are optional. >EXAMPLE< For I := 1 To 10 Do Print(I^2,' '); EndFor; 1 4 9 16 25 36 49 64 81 100 ------------------------------- For I := 1 To 3 Do PrintLn(I); EndFor; 1 2 3 ------------------------------- Print 'hello',NewLine,'world'; hello world ------------------------------- See also: Print On
Product, Sum
SYNTAX Product(L:List):OBJECT Sum(L:List):OBJECT SUMMARY the product or sum of the elements of a list DESCRIPTION These functions return the product and sum, respectively, of the objects in the list L. >EXAMPLE< Use R ::= Q[x,y]; Product([3,x,y^2]); 3xy^2 ------------------------------- Sum([3,x,y^2]); y^2 + x + 3 ------------------------------- Product(1..40)=Fact(40); TRUE ------------------------------- Sum(['c','oc','oa']); cocoa ------------------------------- See also: Algebraic Operators
Quit
SYNTAX Quit SUMMARY quit CoCoA DESCRIPTION This command is used to quit CoCoA. Note, it is issued as follows: Quit; without parentheses. See also: Ciao
QuotientBasis
SYNTAX QuotientBasis(I:IDEAL):LIST SUMMARY vector space basis for zero-dimensional quotient rings DESCRIPTION This function determines a vector space basis (of power products) for the quotient space associated to a zero-dimensional ideal. That is, if R is a polynomial ring with field of coefficients k, and I is a zero-dimensional ideal in R then QuotientBasis(I) is a set of power products forming a k-vector space basis of R/I. The actual set of power products chosen depends on the term ordering in the ring R: the power products chosen are those not divisible by the leading term of any member of the reduced Groebner basis of I. >EXAMPLE< Points:=[[Rand(-9,9) | N In 1..3] | S In 1..25]; Use Q[x,y,z]; I:=IdealOfPoints(Points); QuotientBasis(I); -- power products underneath the DegRevLex reduced GBasis [1, z, z^2, z^3, z^4, y, yz, yz^2, yz^3, y^2, y^2z, y^2z^2, y^3, x, xz, xz^2, xz^3, xy, xyz, xyz^2, xy^2, x^2, x^2z, x^2y, x^3] ------------------------------- Use Q[x,y,z],Lex; I:=IdealOfPoints(Points); QuotientBasis(I); -- power products underneath the Lex reduced GBasis [1, z, z^2, z^3, z^4, z^5, z^6, z^7, z^8, z^9, z^10, z^11, z^12, z^13, y, yz, yz^2, yz^3, yz^4, yz^5, yz^6, y^2, y^2z, y^2z^2, y^2z^3] ------------------------------- See also: IdealOfPoints
QZP, ZPQ
SYNTAX
QZP(F:POLY):POLY
ZPQ(F:POLY):POLY
QZP(F:LIST of POLY):LIST of POLY
ZPQ(F:LIST of POLY):LIST of POLY
QZP(I:IDEAL):IDEAL
ZPQ(I:IDEAL):IDEAL
SUMMARY
change field for polynomials and ideals
DESCRIPTION
These functions map polynomials and ideals of other rings into ones
of the current ring. When mapping from one ring to another, one of
the rings must have coefficients in the rational numbers and the other
must have coefficients in a finite field. The indeterminates in both
rings must be identical.
The function QZP maps polynomials with rational coefficients to
polynomials with coefficients in a finite field; the function ZPQ
does the reverse, mapping a polynomial with finite field coefficients
into one with rational (actually, integer) coefficients. The function
ZPQ is not uniquely defined mathematically, and currently for each
coefficient the least non-negative equivalent integer is chosen.
Users should not rely on this choice, though any change will be
documented.
>EXAMPLE<
Use R::=Q[x,y,z];
F:=1/2*x^3+34/567*x*y*z-890; -- a poly with rational coefficients
Use S::=Z/(101)[x,y,z];
QZP(F); -- compute its image with coeffs in Z/(101)
-50x^3 - 19xyz + 19
-------------------------------
G:=It;
Use R;
ZPQ(G); -- now map that result back to Q[x,y,z]
-- it is NOT the same as F...
51x^3 + 82xyz + 19
-------------------------------
H:=It;
F-H; -- ... but the difference is divisible by 101
-101/2x^3 - 46460/567xyz - 909
-------------------------------
Use S;
QZP(H)-G; -- F and H have the same image in Z/(101)[x,y,z]
0
-------------------------------
See also:
Accessing Other Rings
BringIn
Image
Ring Mappings: the Image Function
Radical
SYNTAX Radical(I:IDEAL):IDEAL SUMMARY radical of an ideal DESCRIPTION This function computes the radical of I using the algorithm described in the paper M. Caboara, P.Conti and C. Traverso: "Yet Another Ideal Decomposition Algorithm." Proc. AAECC-12, pp 39-54, 1997, Lecture Notes in Computer Science, n.1255 Springer-Verlag. NOTE: at the moment, this implementation works only if the coefficient ring is the rationals or has large enough characteristic. >EXAMPLE< Use R ::= Q[x,y]; I := Ideal(x,y)^3; Radical(I); Ideal(y, x) ------------------------------- See also: EquiIsoDec RadicalOfUnmixed
RadicalOfUnmixed
SYNTAX RadicalOfUnmixed(I:IDEAL):IDEAL SUMMARY radical of an unmixed ideal DESCRIPTION This function computes the radical of an unmixed ideal. NOTE: at the moment, this implementation works only if the coefficient ring is the rationals or has large enough characteristic. >EXAMPLE< Use R ::= Q[x,y]; I := Ideal(x^2 - y^2 - 4x + 4y, x - 2); ------------------------------- RadicalOfUnmixed(I); Ideal(x^2 - y^2 - 4x + 4y, x - 2, y - 2) ------------------------------- Minimalized(It); -- the result may not be presented in its simplest form Ideal(x - 2, y - 2) ------------------------------- See also: EquiIsoDec Radical
Rand
SYNTAX Rand():INT Rand(X:INT,Y:INT):INT SUMMARY random integer DESCRIPTION In the first form, the function returns a random integer. In the second, it returns a random integer between X and Y, inclusive. (Note: |X-Y| should be less than 2^33 to assure a more random distribution.) >EXAMPLE< Rand(); 6304433354 ------------------------------- Rand(1,100); 8 ------------------------------- Rand(100,1); 14 ------------------------------- Rand(-10^4,0); -2747 ------------------------------- See also: Integers Randomize Randomized Seed
Randomize, Randomized
SYNTAX Randomize(V:POLY):POLY Randomized(F:POLY or INT):POLY or INT where V is a variable containing a polynomial. SUMMARY randomize the coefficients of a given polynomial DESCRIPTION The first function replaces the coefficients of terms of the polynomial contained in V with randomly generated coefficients. The result is stored in V, overwriting the original polynomial. The second function with a polynomial argument returns a polynomial obtained by replacing the coefficients of F with randomly generated coefficients. The original polynomial, F, is unaffected. With an integer argument, the second function returns a random integer. Note: It is possible that some coefficients will be replaced by zeroes, i.e., some terms from the original polynomial may disappear in the result. >EXAMPLE< Use R ::= Q[x]; F := 1+x+x^2; Randomized(F); -2917104644x^2 + 3623608766x - 2302822308 ------------------------------- F; x^2 + x + 1 ------------------------------- Randomize(F); F; -1010266662x^2 + 1923761602x - 4065654277 ------------------------------- Randomized(23); -3997312402 ------------------------------- Use R ::= Z/(7)[x,y]; Randomized(x^2+3x-5); 3x^2 + 2x - 2 ------------------------------- See also: Rand
Rank
SYNTAX Rank(M:MODULE):INT Rank(M:MAT):INT SUMMARY rank of a module DESCRIPTION This function computes the rank of M. For a module M this is defined as the vector space dimension of the subspace generated by the generators of M over the quotient field of the base ring -- contrast this with the function NumComps which simply counts the number of components the module has. >EXAMPLE< Use R ::= Q[x,y,z]; Rank(Module([x,y,z,0])); 1 ------------------------------- Rank(Module([[1,2,3],[2,4,6]])); 1 ------------------------------- Rank(Module([[1,2,3],[2,5,6]])); 2 -------------------------------
Record
SYNTAX Record[X_1 = OBJECT,...,X_n = OBJECT] where each X_i is a variable. SUMMARY create a record DESCRIPTION This function returns a record with fields "X_1",...,"X_n". The empty record is given by 'Record[]'. The records are "open" in the sense that new fields may be added after the record is first defined. >EXAMPLE< P := Record[ Height = 10, Width = 5]; P.Height * P.Width; 50 ------------------------------- P.Area := It; P; Record[Area = 50, Height = 10, Width = 5] ------------------------------- See also: Fields
ReducedGBasis
SYNTAX
ReducedGBasis(M:IDEAL, MODULE, or TAGGED("Quotient")):LIST
SUMMARY
compute a reduced Groebner basis
DESCRIPTION
If M is an ideal or module, this function returns a list whose
components form a reduced Groebner basis for M with respect to the
term-ordering of the current ring. If M is a quotient of the current
ring by an ideal I or of a free module by a submodule N, then the
Groebner basis for M is defined to be that of I or N, respectively.
>EXAMPLE<
Use R ::= Q[t,x,y,z];
I := Ideal(t^3-x,t^4-y,t^5-z);
GB.Start_GBasis(I); -- start the Interactive Groebner Framework
GB.Step(I); -- take one step towards computing the Groebner basis
I.GBasis; -- the Groebner basis so far
[t^3 - x]
-------------------------------
GB.Complete(I); -- finish the computation
I.GBasis;
[t^3 - x, -tx + y, -ty + z, -y^2 + xz, -x^2 + tz, t^2z - xy]
-------------------------------
ReducedGBasis(I);
[t^3 - x, tx - y, ty - z, y^2 - xz, x^2 - tz, t^2z - xy]
-------------------------------
See also:
Introduction to Groebner Bases in CoCoA
The Interactive Groebner Framework
Repeat
SYNTAX
Repeat C Until B
Repeat C EndRepeat
where C is a sequence of commands and B is a boolean expression.
SUMMARY
loop command
DESCRIPTION
In the first form, the command sequence C is repeated until B
evaluates to FALSE. Unlike the 'While' command, C is executed at
least once. Note that there is no 'EndRepeat' following B. In the
second form, the command sequence C is repeated until a 'Break' or
'Return' is encountered within C.
>EXAMPLE<
Define GCD_Euclid(A,B)
Repeat
R := Mod(A,B);
A := B;
B := R;
Until B = 0;
Return A
EndDefine;
GCD_Euclid(6,15);
3
-------------------------------
N := 0;
Repeat
N := N+1;
PrintLn(N);
Return If N = 5;
EndRepeat;
1
2
3
4
5
-------------------------------
See also:
For
Foreach
While
Res
SYNTAX
Res(M):TAGGED("$cocoa/gb.Res")
where M is of type IDEAL or MODULE or TAGGED("Quotient").
SUMMARY
free resolution
DESCRIPTION
This function returns the minimal free resolution of M. If M is a
quotient of the current ring by an ideal I or a quotient of a free
module by a submodule N, then the resolution of M is defined
to be that of I or N, respectively.
'Res' only works in the homogeneous context, and the coefficient ring
must be a field.
>EXAMPLE<
Use R ::= Q[x,y,z];
I := Ideal(x,y,z^2);
Res(R/I);
0 --> R(-4) --> R(-2)(+)R^2(-3) --> R^2(-1)(+)R(-2) --> R
-------------------------------
Describe It;
Mat[
[y, x, z^2]
]
Mat[
[x, z^2, 0],
[-y, 0, z^2],
[0, -y, -x]
]
Mat[
[z^2],
[-x],
[y]
]
-------------------------------
For fine control and monitoring of Groebner basis calculations,
including various types of trunctations, see 'The Interactive Groebner
Framework' and 'Introduction to Panels'.
See also:
Example: Interactive Resolution Computation
Example: Truncations
Introduction to Groebner Bases in CoCoA
Resultant
SYNTAX Resultant(F:POLY,G:POLY,X:INDET):POLY SUMMARY the resultant of two polynomials DESCRIPTION This function returns the resultant of the polynomials F and G with respect to the indeterminate X. >EXAMPLE< Use R ::= Q[p,q,x]; F := x^3+px-q; G := Der(F,x); Resultant(F,G,x); 4p^3 + 27q^2 ------------------------------- See also: Discriminant Sylvester
Reset
SYNTAX Reset():NULL SUMMARY reset panels and random number seed to defaults DESCRIPTION This function resets the options in the CoCoA panels to their default values and executes 'Seed(0)'. >EXAMPLE< Set Indentation; Panel(GENERAL); Echo............... : FALSE Timer.............. : FALSE Trace.............. : FALSE Indentation........ : TRUE TraceSources....... : FALSE SuppressWarnings... : FALSE ComputationStack... : FALSE ------------------------------- Reset(); Panel(GENERAL); Echo............... : FALSE Timer.............. : FALSE Trace.............. : FALSE Indentation........ : FALSE TraceSources....... : FALSE SuppressWarnings... : FALSE ComputationStack... : FALSE ------------------------------- See also: Introduction to Panels ResetPanels Seed
ResetPanels
SYNTAX ResetPanels():NULL SUMMARY reset panels to their default values DESCRIPTION This function resets the options in the CoCoA panels to their default values. >EXAMPLE< Set Indentation; Panel(GENERAL); Echo............... : FALSE Timer.............. : FALSE Trace.............. : FALSE Indentation........ : TRUE TraceSources....... : FALSE SuppressWarnings... : FALSE ComputationStack... : FALSE ------------------------------- ResetPanels(); Panel(GENERAL); Echo............... : FALSE Timer.............. : FALSE Trace.............. : FALSE Indentation........ : FALSE TraceSources....... : FALSE SuppressWarnings... : FALSE ComputationStack... : FALSE ------------------------------- See also: Introduction to Panels Reset
Return
SYNTAX
Return
Return E
where E is an expression.
SUMMARY
exit from a structured command
DESCRIPTION
This function is used to exit from a structured command. The latter
form returns the value of the expression E to the user. If executed
within a nested loop or inside a user-defined function, it breaks out
back to the "top level", not just to the next higher loop. (For
breaks to the next higher loop, see 'Break'.)
>EXAMPLE<
Define Rev(L) -- reverse a list
If Len(L) < 2 Then Return L EndIf;
M := Rev(Tail(L)); -- recursive function call
H := Head(L);
Return Concat(M,[H]);
EndDefine;
Rev([1,2,3,4];
[4, 3, 2, 1]
-------------------------------
-- Another example: returning from a nested loop
For I := 1 To 5 Do
For J := 1 To 5 Do
If J > 2 Then Return Else Print([I,J],' ') EndIf
EndFor;
EndFor;
[1, 1] [1, 2]
-------------------------------
See also:
Break
Define
Reverse, Reversed
SYNTAX Reverse(V:LIST):NULL Reversed(L:LIST):NULL where V is a variable containing a list in the first case. SUMMARY reverse a list DESCRIPTION The the function 'Reverse' reverses the order of the elements of the list in V, returning Null. It does *not* return the reversed list, but instead changes L itself. The function 'Reversed' returns the reversed list without changing L. >EXAMPLE< L := [1,2,3,4]; Reverse(L); L; -- L has been modified [4, 3, 2, 1] ------------------------------- M := [1,2,3,4]; Reversed(M); -- the reversed list is returned [4, 3, 2, 1] ------------------------------- M; -- M has not been modified [1, 2, 3, 4] ------------------------------- See also: Sort Sorted
Ring
SYNTAX
Ring(R:RING):RING
SUMMARY
returns the ring with a given name
DESCRIPTION
This function returns the ring with identifier R.
>EXAMPLE<
Use R ::= Q[x,y,z];
S ::= Z/(3)[a,b];
Ring(S);
Z/(3)[a,b]
-------------------------------
Ring(R);
Q[x,y,z]
-------------------------------
R; -- same as above, as long as there is no variable with identifier
-- R in the working memory
Q[x,y,z]
-------------------------------
CurrentRing();
Q[x,y,z]
-------------------------------
R := 5; -- a variable with identifier R; now there are two objects
-- with the identifier R: a variable and a ring
R;
5
-------------------------------
Memory(); -- the variables of the working memory
["It", "R"]
-------------------------------
RingEnvs(); -- the list of rings
["Q", "Qt", "R", "S", "Z"]
-------------------------------
Ring(R); -- the ring with identifier R
Q[x,y,z]
-------------------------------
See also:
CurrentRing
Introduction to Rings
RingEnv
RingEnvs
RingEnv
SYNTAX
RingEnv():STRING
RingEnv(E:POLY, IDEAL, MODULE, RATFUN, VECTOR):STRING
SUMMARY
name of the current ring
DESCRIPTION
The first form of this function returns the identifier for the
current ring. The second form returns the identifier of the ring on
which the object E is dependent.
>EXAMPLE<
Use R ::= Q[x,y,z];
I := Ideal(x,y); -- an object dependent on R
S ::= Z/(3)[a,b]; -- define S, but do not make S active
RingEnv(); -- the current ring
R
-------------------------------
RingEnvs(); -- your result here could be different
["Q", "Qt", "R", "S", "Z"]
-------------------------------
Ring(S);
Z/(3)[a,b]
-------------------------------
I;
Ideal(x, y)
-------------------------------
Use S; -- S is now the active ring
I; -- I is labeled by its ring. The label appears explicitly when R
-- is not the current ring.
R :: Ideal(x, y)
-------------------------------
RingEnv(I); -- the ring labeling I
R
-------------------------------
CurrentRing();
Q[x,y,z]
-------------------------------
Use Q[ab];
RingEnv();
CurrentRingEnv
-------------------------------
See also:
CurrentRing
Ring
RingEnvs
RingEnvs
SYNTAX RingEnvs():TAGGED SUMMARY names of all defined rings DESCRIPTION This function returns the tagged list of identifiers for the existing rings. >EXAMPLE< R_1 ::= Q[a,b,c]; R_2 ::= Q[x,y]; RingEnvs(); -- your result may be different ["Q", "Qt", "R", "R_1", "R_2", "Z"] ------------------------------- See also: CurrentRing Ring RingEnv
Saturation, HSaturation
SYNTAX
Saturation(I:IDEAL,J:IDEAL):IDEAL
HSaturation(I:IDEAL,J:IDEAL):IDEAL
SUMMARY
saturation of ideals
DESCRIPTION
These functions return the saturation of I with respect to J: the
ideal of polynomials F such that FG is in I for all G in J^d for some
positive integer d.
The function 'HSaturation' calculates the saturation using a
Hilbert-driven algorithm. It differs from 'Saturation' only when the
input is inhomogeneous, in which case, 'HSaturation' may be faster.
The coefficient ring must be a field.
>EXAMPLE<
Use R ::= Q[xyz];
I := Ideal(x-z,y-2z);
J := Ideal(x-2z,y-z);
K := Intersection(I,J); -- ideal of two points in the
-- projective plane
L := Intersection(K,Ideal(x,y,z)^3); -- add an irrelevant component
Hilbert(R/L);
H(0) = 1
H(1) = 3
H(2) = 6
H(t) = 2 for t >= 3
-------------------------------
Saturation(L,Ideal(x,y,z)) = K; -- saturating gets rid of the
-- irrelevant component
TRUE
-------------------------------
See also:
Colon, :, HColon
ScalarProduct
SYNTAX ScalarProduct(L, M):OBJECT where each of L and M is of type VECTOR or LIST SUMMARY scalar product DESCRIPTION This function returns the sum of the product of the components of L and M; precisely: ScalarProduct(L,M) = Sum([L[I]*M[I]|I In 1..Min(Len(L),Len(M))])). Thus, the function works even if the lengths of L and M are different. The function works whenever the product of the components of L and M are defined (see 'Algebraic Operators'). >EXAMPLE< ScalarProduct([1,2,3],[5,0,-1]); 2 ------------------------------- ScalarProduct([1,2,3],[5,0]); 5 ------------------------------- Use R ::= Q[x,y]; ScalarProduct([Ideal(x,y),Ideal(x^2-xy)],[x^2,y]); Ideal(x^3, x^2y, x^2y - xy^2) ------------------------------- See also: Algebraic Operators
Seed
SYNTAX Seed(N:INT):INT SUMMARY seed for 'Rand' DESCRIPTION This function seeds the random number generator, 'Rand'. >EXAMPLE< Seed(5); Rand(); 732175462471183822614941902 ------------------------------- Rand(); 56676524785038889475475113449 ------------------------------- Seed(5); -- with the same seed, 'Rand' generates the same sequence Rand(); 732175462471183822614941902 ------------------------------- Rand(); 56676524785038889475475113449 ------------------------------- -- The following shows how to make a seed based on the date. D := Date(); D; Mon Mar 02 14:43:44 1998 ------------------------------- F := Sum([Ascii(D[N])| N In 1..Len(D)]); F; [1455] ------------------------------- Seed(F[1]); See also: Rand
SeparatorsOfPoints
SYNTAX
SeparatorsOfPoints(Points:LIST):LIST
where Points is a list of lists of coefficients representing a set of
*distinct* points in affine space.
SUMMARY
separators for affine points
DESCRIPTION
This function computes separators for the points: that is, for each
point a polynomial is determined whose value is 1 at that point and 0
at all the others. The separators yielded are reduced with respect to
the reduced Groebner basis which would be found by 'IdealOfPoints'.
NOTE:
* the current ring must have at least as many indeterminates as the
dimension of the space in which the points lie;
* the base field for the space in which the points lie is taken to be
the coefficient ring, which should be a field;
* in the polynomials returned the first coordinate in the space is
taken to correspond to the first indeterminate, the second to the
second, and so on;
* the separators are in the same order as the points (i.e. the first
separator is the one corresponding the first point, and so on);
* if the number of points is large, say 100 or more, the returned
value can be very large. To avoid possible problems when printing
such values as a single item we recommend printing out the elements
one at a time as in this example:
S:=SeparatorsOfPoints(Pts);
Foreach Element In S Do
PrintLn Element;
EndForeach;
For separators of points in projective space, see
'SeparatorsOfProjectivePoints'.
>EXAMPLE<
Use R ::= Q[x,y];
Points := [[1, 2], [3, 4], [5, 6]];
S := SeparatorsOfPoints(Points); -- compute the separators
S;
[1/8y^2 - 5/4y + 3, -1/4y^2 + 2y - 3, 1/8y^2 - 3/4y + 1]
-------------------------------
[[Eval(F, P) | P In Points] | F In S]; -- verify separators
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
-------------------------------
See also:
GenericPoints
IdealAndSeparatorsOfPoints
IdealAndSeparatorsOfProjectivePoints
IdealOfPoints
IdealOfProjectivePoints
Interpolate
SeparatorsOfProjectivePoints
SeparatorsOfProjectivePoints
SYNTAX
SeparatorsOfProjectivePoints(Points:LIST):LIST
where Points is a list of lists of coefficients representing a set of
*distinct* points in projective space.
SUMMARY
separators for projective points
DESCRIPTION
This function computes separators for the points: that is, for each
point a homogeneous polynomial is determined whose value is non-zero at
that point and zero at all the others. (Actually, choosing the values
listed in Points as representatives for the homogeneous coordinates of
the corresponding points in projective space, the non-zero value will
be 1.) The separators yielded are reduced with respect to the reduced
Groebner basis which would be found by 'IdealOfProjectivePoints'.
NOTE:
* the current ring must have at least one more indeterminate than the
dimension of the projective space in which the points lie, i.e, at
least as many indeterminates as the length of an element of
the input, Points;
* the base field for the space in which the points lie is taken to be
the coefficient ring, which should be a field;
* in the polynomials returned the first coordinate in the space is
taken to correspond to the first indeterminate, the second to the
second, and so on;
* the separators are in the same order as the points (i.e. the first
separator is the one corresponding the first point, and so on);
* if the number of points is large, say 100 or more, the returned
value can be very large. To avoid possible problems when printing
such values as a single item we recommend printing out the elements
one at a time as in this example:
S:=SeparatorsOfProjectivePoints(Pts);
Foreach Element In S Do
PrintLn Element;
EndForeach;
For separators of points in affine space, see
'SeparatorsOfPoints'.
>EXAMPLE<
Use R ::= Q[x,y,z];
Points := [[0,0,1],[1/2,1,1],[0,1,0]];
S := SeparatorsOfProjectivePoints(Points);
S;
[-2x + z, 2x, -2x + y]
-------------------------------
[[Eval(F, P) | P In Points] | F In S]; -- verify separators
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
-------------------------------
See also:
GenericPoints
IdealAndSeparatorsOfPoints
IdealAndSeparatorsOfProjectivePoints
IdealOfPoints
IdealOfProjectivePoints
Interpolate
SeparatorsOfPoints
Set
SYNTAX Set(L:LIST):LIST SUMMARY remove duplicates from a list DESCRIPTION This function returns a list obtained by removing duplicates from L. >EXAMPLE< Set([2,2,2,1,2,1,1,3,3]); [2, 1, 3] ------------------------------- NOTE: there is a command named 'Set' for setting panel options. It's search key in online help is 'Set, Unset'. The search key for the present function is 'Set'. See also: EqSet Intersection, IntersectionList Remove
Set, Unset
SYNTAX Set O Set O := B:BOOL UnSet O where O is a panel option. SUMMARY set and unset panel options DESCRIPTION The command 'Set' in its first form sets a panel option to TRUE. The command 'UnSet' sets a panel option to FALSE. The command 'Set' in the second-listed form can be used to set an option to TRUE or FALSE. A list of panels is returned by 'Panels()', and a list of panel options for a panel with name P is printed by 'Panel(P)'. The current status of an option is returned by 'Option'. >EXAMPLE< Panel(GROEBNER); Sugar........... : TRUE FullRed......... : TRUE SingleStepRed... : FALSE Verbose......... : FALSE ------------------------------- Set Verbose; UnSet Sugar; Set FullRed := FALSE; Panel(GROEBNER); Sugar........... : FALSE FullRed......... : FALSE SingleStepRed... : FALSE Verbose......... : TRUE ------------------------------- NOTE: there is also a function called 'Set' which takes a list obtained by removing duplicate elements. The search key for that function in online help is 'Set' and the search key for the present command is 'Set, Unset'. See also: Introduction to Panels Panel Option Panels
Shape
SYNTAX
Shape(E:LIST):LIST (of TYPE)
Shape(E:MAT):MAT (of TYPE)
Shape(E:RECORD):RECORD (of TYPE)
Shape(E:OTHER):TYPE
where OTHER stands for a type which is not LIST, MAT, or RECORD.
SUMMARY
extended list of types involved in an expression
DESCRIPTION
This function returns the extended list of types involved in the
expression E as outlined below:
Type(E) = LIST
In this case, Shape(E) is the list whose i-th component is the type
of the i-th component of E.
Type(E) = MAT
In this case, Shape(E) is a matrix with (i,j)-th entry equal to the
type of the (i,j)-th entry of E.
Type(E) = RECORD
In this case, Shape(E) is a record whose fields are the types of the
fields of E.
Otherwise, Shape(E) is the type of E.
>EXAMPLE<
Use R ::= Q[x];
L := [1,[1,"a"],x^2-x];
Shape(L);
[INT, [INT, STRING], POLY]
-------------------------------
R := Record(Name = 'test', Contents = L);
Shape(R);
Record[Contents = [INT, [INT, STRING], POLY], Name = STRING]
-------------------------------
It.Name;
STRING
-------------------------------
There are undocumented functions, 'IsSubShape' and 'IsSubShapeOfSome',
for determining if the 'shape' of an CoCoA expression is a 'subshape'
of another. To see the code for these functions, enter
'Describe Function('$cocoa/misc.IsSubShape')' or
'Describe Function('$cocoa/misc.IsSubShapeOfSome')'.
See also:
Data Types
Size
SYNTAX Size(E:OBJECT):INT SUMMARY the amount of memory used by an object DESCRIPTION This function returns the amount of memory used by the object E, expressed in words (1 word = 4 bytes = 32 bits). >EXAMPLE< Use R ::= Q[x,y]; Size(1); 1 ------------------------------- Size(2^32-1); 1 ------------------------------- Size(2^32); 2 ------------------------------- Size(2^64); 3 ------------------------------- Size(x); 32 ------------------------------- Size([x,y]); 64 ------------------------------- See also: Count Len
Skip
SYNTAX Skip SUMMARY does nothing DESCRIPTION This command does nothing. I suppose it might be used to make the structure of a user-defined function more clear. It is probably at least as useful as the function 'Tao'. >EXAMPLE< Skip;
Sort, Sorted
SYNTAX Sort(V:LIST):NULL Sorted(L:LIST):LIST where V is a variable containing a list. SUMMARY sort a list DESCRIPTION The first function sorts the elements of the list in V with respect to the default comparisons related to their types; it overwrites V. The second function, 'Sorted', returns the list of the sorted elements of L without affecting L, itself. For more on the default comparisions, see 'Relational Operators' in the chapter on operators. For more complicated sorting, see 'SortBy, SortedBy'. >EXAMPLE< L := [3,2,1]; Sort(L); L; [1, 2, 3] ------------------------------- Use R ::= Q[x,y,z]; L := [x,y,z]; Sort(L); L; [z, y, x] ------------------------------- Sorted([y,x,z,x^2]); [z, y, x, x^2] ------------------------------- Sorted([3,1,1,2]); [1, 1, 2, 3] ------------------------------- Sorted(["b","c","a"]); ["a", "b", "c"] ------------------------------- Sorted([Ideal(x,y),Ideal(x)]); -- ideals are ordered by containment [Ideal(x), Ideal(x, y)] ------------------------------- See also: Relational Operators SortBy, SortedBy
SortBy, SortedBy
SYNTAX
SortBy(V:LIST,F:FUNCTION):NULL
SortedBy(L:LIST,F:FUNCTION):LIST
where V is a variable containing a list and F is a boolean-valued
function of two arguments.
SUMMARY
sort a list
DESCRIPTION
The first function sorts the elements of the list in V with respect
to the comparisons made by F; it overwrites V. The second function,
'Sorted', returns the list of the sorted elements of L without
affecting L, itself. The function F takes two arguments and returns
TRUE or FALSE. If F(A,B) returns TRUE, then 'SortBy' and 'SortedBy'
will place A before B in the sorted list.
>EXAMPLE<
Define ByLength(S,T) -- define the sorting function
Return Len(S) > Len(T);
EndDefine;
M := ['dog','mouse','cat'];
SortedBy(M,Function('ByLength'));
["mouse", "dog", "cat"]
-------------------------------
M; -- M is not changed
["dog", "mouse", "cat"]
-------------------------------
Sorted(M); -- the function 'Sort' sorts using the default ordering:
-- in this case, alphabetical order.
["cat", "dog", "mouse"]
-------------------------------
SortBy(M,Function('ByLength')); -- sort M in place, changing M
M;
["mouse", "dog", "cat"]
-------------------------------
See also:
Sort, Sorted
Source, <<
SYNTAX
Source S:STRING
<< S:STRING
SUMMARY
read commands from a file or device
DESCRIPTION
This command executes all CoCoA commands in the file or device named
S. A typical use of 'Source' is to collect user-defined functions and
variables in a text file, say, 'MyFile.coc' and then execute:
Source 'MyFile.coc';
or, equivalently,
<< 'MyFile.coc';
Functions and variables read in from a file in this way will erase
functions and variables with identical names that may already exist.
This can be avoided by using packages. Repeatedly used functions can be
read into CoCoA at start-up by using 'Source' in the 'userinit.coc'
file.
See also:
Introduction to IO
Introduction to Packages
User Initialization
Spaces
SYNTAX Spaces(N:INT):STRING SUMMARY return a string of spaces DESCRIPTION This function returns a string consisting of N spaces. >EXAMPLE< L := 'a' + Spaces(5) + 'b'; L; a b ------------------------------- See also: Dashes Equals
Sprint
SYNTAX Sprint(E:OBJECT):STRING SUMMARY convert to a string DESCRIPTION This function takes any CoCoA expression and converts its value to a string. One use is to check for extremely long output before printing in a CoCoA window. >EXAMPLE< Use R ::= Q[x,y]; I := Ideal(x,y); J := Sprint(I); I; Ideal(x, y) ------------------------------- J; -- The output for I and J looks the same, but ... Ideal(x, y) ------------------------------- Type(I); -- I is an ideal, and IDEAL ------------------------------- Type(J); -- J is just the string 'Ideal(x, y)'. STRING ------------------------------- J[1]; -- the 1st character of J I ------------------------------- J[2]; -- the 2nd character of J d ------------------------------- Len(J); -- J has 11 characters 11 ------------------------------- See also: Introduction to IO IO.SprintTrunc Print, PrintLn
StarPrint
SYNTAX StarPrint(F:POLY):NULL SUMMARY print polynomial with *'s for multiplications DESCRIPTION This function prints the polynomial F with asterisks added to denote multiplications. This function may be useful when cutting and pasting from CoCoA to other mathematical software. >EXAMPLE< Use R ::= Q[x,y]; F := x^3+2xy-y^2; StarPrint(F); 1*x^3+2*x*y-1*y^2 -------------------------------
Starting
SYNTAX
Starting(S:STRING):LIST of STRING
SUMMARY
list functions starting with a given string
DESCRIPTION
This function returns a list of all CoCoA functions starting with the
string 'S'. In general, this list will include undocumented
commands. For these, one may hope to find more information using
'Describe Function('FunctionName')' or 'Describe
Function('$PackageName.FunctionName')'.
>EXAMPLE<
Starting('Su');
["SubstPoly", "SubSet", "Sum", "Subst", "Support"]
-------------------------------
See also:
Other Help
Submat
SYNTAX Submat(M:LIST or MAT,R:LIST of INT,C:LIST of INT):MAT SUMMARY submatrix DESCRIPTION This function returns the submatrix of M formed by the rows listed in R and the columns listed in C. If M is a list, it is interpreted as a matrix in the natural way. >EXAMPLE< M := Mat([[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]); Submat(M,[1,3],3..5); Mat[ [3, 4, 5], [13, 14, 15] ] ------------------------------- L := [[1,2,3],[4,5,6]]; Submat(L,[2],[1,3]); Mat[ [4, 6] ] ------------------------------- See also: Introduction to Matrices Minors
SubSet
SYNTAX SubSet(L:LIST,M:LIST):BOOL SUMMARY checks if the elements of one list are a subset of another DESCRIPTION This function returns TRUE is Set(L) is contained in Set(M); otherwise it returns FALSE. >EXAMPLE< SubSet([1,1,2],[1,2,3,"a"]); TRUE ------------------------------- SubSet([1,2],["a","b"]); FALSE ------------------------------- SubSet([],[1,2]); TRUE ------------------------------- See also: EqSet Set
Subst
SYNTAX Subst(E:OBJECT,X,F):OBJECT Subst(E:OBJECT,[[X_1,F_1],...,[X_r,F_r]]):OBJECT where each X or X_i is an indeterminate and each F or F_i is a number, polynomial, or rational function. SUMMARY substitute values for indeterminates DESCRIPTION The first form of this function substitutes F_i for X_i in the expression E. The second form is a shorthand for the first in the case of a single indeterminate. When substituting for the indeterminates in order, it is easier to use 'Eval'. >EXAMPLE< Use R ::= Q[x,y,z,t]; F := x+y+z+t^2; Subst(F,x,-2); t^2 + y + z - 2 ------------------------------- Subst(F,x,z/y); (yt^2 + y^2 + yz + z)/y ------------------------------- Subst(F,[[x,x^2],[y,y^3],[z,t^5]]); t^5 + y^3 + x^2 + t^2 ------------------------------- Eval(F,[x^2,y^3,t^5]); -- the same thing as above t^5 + y^3 + x^2 + t^2 ------------------------------- MySubst := [[y,1],[t,3z-x]]; Subst(xyzt,MySubst); -- substitute into the function xyzt -x^2z + 3xz^2 ------------------------------- See also: Eval Evaluation of Polynomials Image QZP, ZPQ Substitutions
Support
SYNTAX Support(F:POLY or VECTOR):LIST SUMMARY the list of terms of a polynomial or vector DESCRIPTION This function returns the list of terms of F. To get a list of monomials, which includes coefficients, use 'Monomials'. >EXAMPLE< Use R ::= Q[x,y]; F := 3x^2-4xy+y^3+3; Support(F); [y^3, x^2, xy, 1] ------------------------------- Monomials(F); [y^3, 3x^2, -4xy, 3] ------------------------------- Support(Vector(x^2y,x^3-3y^2,34)); [Vector(0, x^3, 0), Vector(x^2y, 0, 0), Vector(0, y^2, 0), Vector(0, 0, 1)] ------------------------------- See also: Coefficients Monomials
Sylvester
SYNTAX Sylvester(F:POLY,G:POLY,X:INDET) SUMMARY the Sylvester matrix of two polynomials DESCRIPTION This function returns the Sylvester matrix of the polynomials F and G with respect to the indeterminate X. This is the matrix used to calculate the resultant. >EXAMPLE< Use R ::= Q[p,q,x]; F := x^3+px-q; G := Der(F,x); Sylvester(F,G,x); Mat[ [1, 0, p, -q, 0], [0, 1, 0, p, -q], [3, 0, p, 0, 0], [0, 3, 0, p, 0], [0, 0, 3, 0, p] ] ------------------------------- Det(Sylvester(F,G,x)) = Resultant(F,G,x); TRUE ------------------------------- See also: Resultant
Syz
SYNTAX Syz(L:LIST of POLY):MODULE Syz(L:LIST of VECTOR):MODULE Syz(M:IDEAL or MODULE, Index:INT):MODULE SUMMARY syzygy modules DESCRIPTION In the first two forms this function computes the syzygy module of a list of polynomials or vectors. In the last form this function returns the specified syzygy module of the minimal free resolution of M which must be homogeneous. As a side effect, it computes the Groebner basis of M. The coefficient ring must be a field. >EXAMPLE< Use R ::= Q[xyz]; Syz([x^2-y,xy-z,xy]); Module([0, xy, -xy + z], [z, x^2 - y, -x^2 + y], [yz, -y^2, y^2 - xz], [xy, 0, -x^2 + y]) ------------------------------- I := Ideal(x^2-yz, xy-z^2, xyz); Syz(I,0); Module([x^2 - yz], [xy - z^2], [xyz]) ------------------------------- Syz(I,1); Module([-x^2 + yz, xy - z^2, 0], [xz^2, -yz^2, -y^2 + xz], [z^3, 0, -xy + z^2], [0, z^3, -x^2 + yz]) ------------------------------- Syz(I,2); Module([0, z, -x, y], [-z^2, -x, y, -z]) ------------------------------- Syz(I,3); Module([0]) ------------------------------- Res(I); 0 --> R^2(-6) --> R(-4)(+)R^3(-5) --> R^2(-2)(+)R(-3) ------------------------------- For fine control and monitoring of Groebner basis calculations, see 'The Interactive Groebner Framework' and 'Introduction to Panels.' See also: Introduction to Groebner Bases in CoCoA
SyzMinGens
SYNTAX SyzMinGens: FUNCTION ELIMINATED SUMMARY syzygy module for a set of minimal generators DESCRIPTION The SyzMinGens function has been removed. See also: Syz SyzOfGens
SyzOfGens
SYNTAX
SyzOfGens(M:IDEAL, MODULE, or TAGGED("Quotient")):MODULE
SUMMARY
syzygy module for a given set of generators
DESCRIPTION
If M is an ideal or module, this function calculates the syzygy
module for the given set of generators of M. If M is a quotient of
the current ring by an ideal I or a quotient of a free module by a
submodule N, then this function calculates the syzygy module for the
given set of generators of I or N, respectively.
The coefficient ring must be a field.
>EXAMPLE<
Use R ::= Q[x,y];
I := Ideal(x,y,x+y);
SyzOfGens(I);
Module([1, 1, -1], [y, -x, 0])
-------------------------------
See also:
Syz
SyzMinGens
Tag
SYNTAX
Tag(E:OBJECT):STRING
SUMMARY
returns the tag string of an object
DESCRIPTION
If E is a tagged object, this function returns the tag of E;
otherwise, it returns the empty string.
>EXAMPLE<
L := Tagged(3,'MyTag');
Type(L);
TAGGED("MyTag")
-------------------------------
Tag(L);
MyTag
-------------------------------
See also:
Tagged Printing
Tagged, Untagged, @
SYNTAX
Tagged(E:OBJECT,S:STRING):TAGGED(S)
Untagged(E:TAGGED_OBJECT):UNTAGGED_OBJECT
@E:TAGGED_OBJECT:UNTAGGED_OBJECT
SUMMARY
tag or untag an object for pretty printing
DESCRIPTION
The first function returns the object E, tagged with the string S.
The second strips E of its tag, if any. The "at sign" can also be
used to untag an object: @E is equivalent to Untagged(E). These
functions are used for pretty printing of objects. See the reference
listed below.
>EXAMPLE<
L := [1,2,3];
M := Tagged(L,'MyTag');
Type(L);
LIST
-------------------------------
Type(M);
TAGGED("MyTag")
-------------------------------
Type(Untagged(M));
LIST
-------------------------------
Type(@M);
LIST
-------------------------------
See also:
Tagged Printing
Tail
SYNTAX Tail(L:LIST):OBJECT SUMMARY remove the first element of a list DESCRIPTION This function returns the list obtained from L by removing its first element. It cannot be applied to the empty list. >EXAMPLE< Tail([1,2,3]); [2, 3] ------------------------------- See also: First Head Last
TensorMat
SYNTAX TensorMat(M:Mat, N:Mat):MAT SUMMARY returns the tensor product of two matrices DESCRIPTION This function returns the tensor product of two matrices. >EXAMPLE< Use R ::= Q[x,y,z,w]; TensorMat(Mat([[1,-1],[2,-2],[3,-3]]),Mat([[x,y],[z,w]])); Mat[ [x, y, -x, -y], [z, w, -z, -w], [2x, 2y, -2x, -2y], [2z, 2w, -2z, -2w], [3x, 3y, -3x, -3y], [3z, 3w, -3z, -3w] ] -------------------------------
Toric
SYNTAX Toric(L:LIST of BINOMIAL):IDEAL Toric(L:LIST of BINOMIAL,X:LIST of INDETS):IDEAL Toric(M:MAT or LIST of LIST):IDEAL where the entries of M are non-negative. Elements of L must be homogeneous (w.r.t. the first row of the weights matrix). SUMMARY saturate toric ideals DESCRIPTION These functions return the saturation of an ideal, I, generated by binomials. In the first two cases, I is the ideal generated by the binomials in L. To describe the ideal in the last case, let K be the integral elements in the kernel of M. For each k in K, we can write k = k(+) - k(-) where the i-th component of k(+) is the i-th component of k, if positive, otherwise zero. Then I is the ideal generated by the binomials x^k(+) - x^k(-) as k ranges over K. Note: successive calls to this last form of the function may produce different generators for the saturation. The first and third functions return the saturation of I. For the second function, if the saturation of I with respect to the variables in X happens to equal the saturation of I, then the saturation of I is returned. Otherwise, an ideal *containing* the saturation with respect to the given variables is returned. The point is that if one knows, a priori, that the saturation of I can be obtained by saturating with respect to a subset of the variables, the second function may be used to save time. For more details, see the article: A.M. Bigatti, R. La Scala, L. Robbiano, "Computing Toric Ideals," Preprint (1998). The article describes three different algorithms; the one implemented in CoCoA is "EATI". The first two examples below are motivated by B. Sturmfels, "Groebner Bases and Convex Polytopes," Chapter 6, p. 51. They count the number of homogeneous primitive partition identities of degrees 8 and 9. >EXAMPLE< Use Q[x[1..8]y[1..8]]; HPPI8 := [x[1]^I x[I+2] y[2]^(I+1) - y[1]^I y[I+2] x[2]^(I+1) | I In 1..6]; BL := Toric(HPPI8, [x[1],y[2]]); Len(BL.Gens); 340 ------------------------------- Use Q[x[1..9]y[1..9]]; HPPI9 := [x[1]^I x[I+2] y[2]^(I+1) - y[1]^I y[I+2] x[2]^(I+1) | I In 1..7]; BL := Toric(HPPI9, [x[1],y[2]]); Len(BL.Gens); ------------------------------- 798 ------------------------------- Use R ::= Q[x,y,z,w]; Toric(Ideal(xz-y^2, xw-yz)); Ideal(-y^2 + xz, -yz + xw, z^2 - yw) ------------------------------- Toric([xz-y^2, xw-yz]); Ideal(-y^2 + xz, -yz + xw, z^2 - yw) ------------------------------- Toric([xz-y^2, xw-yz], [y]); Ideal(-y^2 + xz, -yz + xw, z^2 - yw) ------------------------------- Use R ::= Q[x,y,z]; Toric([[1,3,2],[3,4,8]]); Ideal(-x^16 + y^2z^5) ------------------------------- Toric(Mat([[1,3,2],[3,4,8]])); Ideal(-x^16 + y^2z^5) ------------------------------- See also: Toric.CheckInput
Toric.CheckInput
SYNTAX
Toric.CheckInput(E:OBJECT):BOOL
Toric.CheckInput(E:OBJECT,X:LIST):BOOL
SUMMARY
check input to 'Toric'
DESCRIPTION
This function checks if E or (E,X) is suitable input for 'Toric'.
Thus, E should be either a list of homogeneous binomials (without
coefficients) or a matrix of non-negative integers. In the former
case, X must be a list of indeterminates (in the latter, X would be
ignored by 'Toric' anyway).
>EXAMPLE<
Use R ::= Q[x,y,z];
Toric.CheckInput([[1,2,3,4],[4,5,6,7]]);
TRUE
-------------------------------
Toric.CheckInput([[-1,2],[3,4]]);
ERROR: entries must be non-negative integers
CONTEXT: Return(Error(Toric_IntMatrix))
-------------------------------
Toric.CheckInput([xy-z^2,x^3-y^2z]);
TRUE
-------------------------------
Toric.CheckInput([3xy-z^2,x^3-y^2z]); -- the binomials should not
-- have coefficients
ERROR: generators must be of type: power-product - power-product
CONTEXT: Return(Error(Toric_PP))
-------------------------------
Toric.CheckInput([xy-z^2,x^3-y^2z],[x]);
TRUE
-------------------------------
See also:
Toric
Transposed
SYNTAX Transposed(M:MAT):MAT SUMMARY the transposition of a matrix DESCRIPTION This function returns the transpose of the matrix M. >EXAMPLE< M := Mat([[1,2,3],[4,5,6]]); M; Mat[ [1, 2, 3], [4, 5, 6] ] ------------------------------- Transposed(M); Mat[ [1, 4], [2, 5], [3, 6] ] -------------------------------
Type
SYNTAX
Type(E:OBJECT):TYPE
SUMMARY
the data type of an expression
DESCRIPTION
This function returns the data type of E. The function 'Types'
returns the list of CoCoA data types.
>EXAMPLE<
Define CollectInts(L)
Result := [];
Foreach X In L Do
If Type(X) = INT Then Append(Result,X) EndIf
EndForeach;
Return Result
EndDefine;
CollectInts([1,'a',2,'b',3,'c']);
[1, 2, 3]
-------------------------------
Type(Type(INT)); -- Type returns a value of type TYPE
TYPE
-------------------------------
Types();
[NULL, BOOL, STRING, TYPE, ERROR, RECORD, DEVICE, INT, RAT, ZMOD,
POLY, RATFUN, VECTOR, IDEAL, MODULE, MAT, LIST, RING, TAGGED("),
FUNCTION]
-------------------------------
See also:
Data Types
Types
TypeOfCoeffs
SYNTAX TypeOfCoeffs():TYPE SUMMARY type of the coefficients of the current ring DESCRIPTION This function returns the type of the coefficients of the current ring. >EXAMPLE< Use R ::= Q[x,y,z]; TypeOfCoeffs(); RAT ------------------------------- Use S ::= Z/(2)[t]; TypeOfCoeffs(); ZMOD ------------------------------- See also: Characteristic Coefficients CurrentRing Indets
Types
SYNTAX
Types()
SUMMARY
lists all data types
DESCRIPTION
This function lists all CoCoA data types.
>EXAMPLE<
Types();
[NULL, BOOL, STRING, TYPE, ERROR, RECORD, DEVICE, INT, RAT, ZMOD,
POLY, RATFUN, VECTOR, IDEAL, MODULE, MAT, LIST, RING, TAGGED("),
FUNCTION]
-------------------------------
See also:
Data Types
Type
Use
SYNTAX Use N where N is either the identifier of an existing ring or a ring itself. SUMMARY command for making a ring active DESCRIPTION 'Use' is the command for making a ring active, i.e. for making a ring the current ring. The command Use N ::= E; where E is a ring, is a shorthand for N ::= E; Use N; >EXAMPLE< Use S ::= Q[x,y,z]; RingEnv(); S ------------------------------- T::= Z/(3)[a,b]; Use T; RingEnv(); T ------------------------------- Use Q[u]; -- note that 'Use' can be used w/out a ring identifier RingEnv(); CurrentRingEnv ------------------------------- CurrentRing(); Q[u] ------------------------------- See also: Accessing Other Rings Using
Using
SYNTAX
Using R Do C EndUsing
where R is the identifier for a ring and C is a sequence of commands.
SUMMARY
perform commands in non-active ring
DESCRIPTION
Suppose S is the current ring and R is another ring, then
Using R Do
C;
EndUsing;
is equivalent to
Use R;
C;
Use S;
>EXAMPLE<
Use S ::= Q[x,y]; -- the current ring is S
R ::= Q[a,b,c]; -- another ring
Using R Do Indets() EndUsing;
[a, b, c]
-------------------------------
Note: 'Using Q[a,b] Do ...' is not proper syntax and will produce an
error.
See also:
Accessing Other Rings
Use
Var
SYNTAX
Var X
Var(X)
Var(S:STRING)
where X is the identifier of a CoCoA variable.
SUMMARY
function calls by reference, other complex referencing
DESCRIPTION
In the first and second form 'Var' is used as a formal parameter to a
user-defined function. It is used to pass a variable---not its
value---to the user-defined function. The following example should
make the difference clear.
>EXAMPLE<
Define CallByRef(Var L ) -- "call by reference": The variable referred
L := 'new value'; -- to by L is changed.
EndDefine;
M := 'old value';
CallByRef(M);
M;
new value
-------------------------------
Define CallByVal(L) -- "call by value": The value of L is passed to
L := 'new value'; -- the function.
Return L;
EndDefine;
L := 'old value';
CallByVal(L);
new value
-------------------------------
L;
old value.
-------------------------------
In the third form, Var(S), references the value of the variable or ring
whose identifier is S:
>EXAMPLE<
Var("a string") := 6;
Var("a string");
6
-------------------------------
P := Record[Name = 'test', Value = 1];
X := 'Name';
P.Var(X);
test
-------------------------------
Var('myring') ::= Q[a,b];
Var('myring');
Q[a,b]
-------------------------------
Using Var('myring') Do (a+b)^2 EndUsing;
a^2 + 2ab + b^2
-------------------------------
See also:
Define
Vector
SYNTAX Vector(F_1:POLY,...,F_n:POLY):VECTOR Vector(L:LIST):VECTOR where L is a list of polynomials. SUMMARY create a vector DESCRIPTION The first form returns the vector with components F_1,...,F_n; the second form returns the vector whose components are the components of the list L. >EXAMPLE< Use R ::= Q[x]; V := Vector(1,x,x^2); Type(V); VECTOR Vector([1+x,x,x^2]); Vector(x + 1, x, x^2) -------------------------------
WeightsList
SYNTAX WeightsList():LIST SUMMARY first row of the weights matrix DESCRIPTION This function returns the first row of the weights matrix for the current ring as a list. >EXAMPLE< Use R ::= Q[t,x,y,z]; WeightsList(); [1, 1, 1, 1] ------------------------------- Use R ::= Q[t,x,y,z], Weights(1,3,5,2); WeightsList(); [1, 3, 5, 2] ------------------------------- Use R ::= Q[x,y], Weights(Mat([[1,2],[3,4],[5,6]])); WeightsList(); [1, 2] ------------------------------- See also: Deg MDeg Weights Modifier WeightsMatrix
WeightsMatrix
SYNTAX WeightsMatrix():MAT SUMMARY matrix of generalized weights for indeterminates DESCRIPTION This function returns the weights matrix for the current ring. >EXAMPLE< Use R ::= Q[x,y], Weights(Mat([[1,2],[3,4],[5,6]])); WeightsMatrix(); Mat[ [1, 2], [3, 4], [5, 6] ] ------------------------------- MDeg(y); [2, 4, 6] ------------------------------- WeightsList(); -- the first row of the weights matrix [1, 2] ------------------------------- See also: Deg MDeg Weights Modifier WeightsList
While
SYNTAX While B Do C EndWhile where B is a boolean expression and C is a sequence of commands. SUMMARY loop command DESCRIPTION The command sequence C is repeated until B evaluates to FALSE. >EXAMPLE< N := 0; While N <= 5 Do PrintLn(2,"^",N," = ",2^N); N := N+1; EndWhile; 2^0 = 1 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 2^5 = 32 ------------------------------- See also: For Foreach Repeat
WithoutNth
SYNTAX WithoutNth(L:LIST,N:INT):NULL SUMMARY removes the N-th component from a list DESCRIPTION This function returns the list obtained by removing the N-th component of the list L. The list L is not affected (as opposed to the command 'Remove'). >EXAMPLE< L := [1,2,3,4,5]; WithoutNth(L,3); [1, 2, 4, 5] ------------------------------- See also: Insert, Remove
WLog
SYNTAX WLog(F:POLY):LIST of INT SUMMARY weighted list of exponents DESCRIPTION This function returns the weighted list of exponents of the leading term of F, as determined by the first row of the weights matrix. Thus, if all the weights are 1, this function returns the same thing as 'Log(F)'. >EXAMPLE< Use R ::= Q[x,y]; F := x^2-y; WLog(F); [2, 0] ------------------------------- Use R ::= Q[x,y],Weights(2,3); F := x^2-y; WLog(F); [4, 0] ------------------------------- See also: Log