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Poincare, HilbertSeries
Syntax
Poincare(M:RING or TAGGED("Quotient")):TAGGED("$cocoa/hp.PSeries")
HilbertSeries(M:RING or TAGGED("Quotient")):TAGGED("$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: