Zbl. 1090.13021
Computational commutative algebra. II. (English)
Berlin: Springer. x, 586 p. EUR 49.95; sFr. 88.50;
GBP 38.50; US$ 64.95 (2005). [ISBN 3-540-25527-3]
The book under review is the second volume of the authors'
`Computational commutative algebra'.
The first part is published in (2000; Zbl 0956.13008).
In the review of the first part, it was written that the second volume
will appear soon. Now it turns out that soon means five years.
While it took three years to complete the three centimeter
thick first volume the authors continued with the five centimeter
of the second volume with the same continuity of speed and
collection of material as they write in the introduction.
In fact, the second volume of their
``Computational commutative algebra" follows the same line of spirit
with an ingenious selection of the topics and with a lot of fun for
the reader and -- as it seems -- also for the authors.
Besides of an Introduction and several appendices these five
centimeters of text are splitted in three main Chapters (4. The
homogeneous case, 5. Hilbert functions, and 6. Further Applications)
by continuing with the counting of the main Chapters
of the first volume.
In the 4. Section the authors present a detailed study about gradings.
While in the first volume there is a general approach of gradings in
order to provide the basics for Gröbner bases of modules it turns out
that gradings on polynomial rings by arbitrary commutative monoids are
too general. The gradings on $P = K[x_0,...,x_n]$, K a field,
investigated here are $\Bbb Z^m$-gradings defined by a matrix
$M \in Mat_{m,n}(\Bbb Z),$ where the gradings of the
variables are given by the columns of $W$.
More restictive they consider gradings of
positive type, i.e. those defined by matrices $W$ such that some linear
combination of its rows with integer coefficients have all entries
positive. By a grading of positive type the vector space dimension of
all elements of any given degree has finite $K$-dimension. Moreover the
graded version of Nakayama's lemma holds for such gradings. This is
extended to a slight variation of the homogeneous Buchberger Algorithm
for getting a minimal homogeneous system of generators contained in the
original system of generators. This is iterated in order to receive a
minimal graded free resolution. This investigation results out of the
authors' consequent point of view of comptutational aspects of graded
modules.
Various proceedures for passing from a non-homogeneous
situation to a homogeneous one are another feature of this chapter.
More precisely they consider submodules of a finitely generated graded
free $P$-module and they introduce the concept of Macaulay bases. To be
more precise, a positive grading induces a ordering on the set of
terms. By mimicing the developments of Volume 1 the lexicographically
largest degree form of a polynomial is called the degree form.
A set of vectors is called a Macaulay basis of a submodule of
a free module if their degree forms generate the degree
form module of the module. This
corresponds to term orderings in Gröbner basis theory. Often the
algorithmic approaches developed can be improved in case the input
polynomials are homogeneous. A large part of the chapter is devoted to
homogenization, in particular effective methods for computing the
homogenization and the behaviour of Gröbner bases of ideals under
dehomogenization.
Section 5 is devoted to the study of Hilbert functions. It is a
well-known fact that the computation of the Hilbert function as well as
the Hilbert series might be done by the reduction to the case of
monomial ideals. This makes the computation fast and in fact
independent on the minimal free resolution of a module. In this chapter
there is a thorough investigation of the subject starting from the
basics, illustrating the importance of the invariants derived of the
Hilbert functions as dimension, regularity index.
The authors continue
with the bounds for Hilbert functions culminating with the Theorems of
Macaulay and Green. Following their intention on homogenization they
introduce the affine Hilbert function. Consequently they prove that the
dimension of an affine algebra defined via the degree of its affine
Hilbert function is equal to its Krull dimension. As a consequence of
the undertaking there are subsections on primary decompositions,
dimension theory and Noether normalization.
While most of the part of
the section is written for the case of usually (i.e. $\Bbb Z$) graded
rings and modules they include a subsetion on the multigraded situation
as it is neeeded for instance in the study of toric varieties. So,
there is an investigation on multivariante Hilbert series. To do this
seriously the authors have to introduce the $\sigma$-Laurent series
ring for a monoid ordering $\sigma$ on $\Bbb Z^m$. As in the case of
$\Bbb Z$-grading the authors present a Hilbert driven Gröbner basis
computation also in the multivariate case. Among the tutorials of the
chapter there are those on Veronese subrings, Rees rings and Segre
products, from the computational point of view available for the first
time in a textbook.
In the final Section 6 ``Further Applications" there is a summary of
separate subjects related to computational commutative algebra and of
some interest in itself. The first of them is devoted to toric ideals
and Hilbert bases. The approach is based on integer matrices and leads
to the computation of lattice ideals and their saturation. As an
application there is a tutorial on magic squares. In order to study
vanishing ideals of projective point sets (subsection 6.3) by liftings
of monomial ideals the subsection 6.2 provides liftings of ideals and
distractions. Here an ideal $J$ is called a lifting of $I$ whenever
$x_0$ is a non-zero divisor on $R/I$ and $J$ is obtained by
substituting $x_0 = 0$ in $I.$ Recently there has been a deep interest
in the study of vanishing ideals of points in projective $n$-space.
Here the reader might found a computational aspect about them regarding
its Hilbert function, the Cayley-Bacharach property, and the minimal
resolution conjecture. Further subsections are centered around the
border bases of zero-dimensional ideals, filtrations, SAGBI bases, and
automatic theorem proving. In particular there is a study of
singularities, the proof of Molien's theorem, and applications in
elementary geometry. In fact the book alltogether covers on its 586
pages a wealth of interesting material with several unexpected
applications.
As in the first part there is a strong recommandation to an interested
reader to experiment with the computer algebra system CoCoA. To this
end there is a Chapter on the ABC of CoCoA introducing new features and
functions that have been added since the first volume appeared.
Moreover there is a Chapter with suggestions for further reading for
each of the sections. Alltogether now there are 99 tutorials including
Chess playing, Photogrammetry, and Error-Correcting codes.
The book is by the same an encyclopedia on computational commutative
algebra, a source for lectures on the subject as well as an inspiration
for seminars. The text is recommanded for all those who want to learn
and enjoy an algebraic tool that becomes more and more relevant to
different fields of applications.
Peter Schenzel (Halle)
Keywords:
Computational algebra; Gröbner bases;
Hilbert functions; homogeneous rings; graded rings