CoCoA:AffHilbertSeries

AffHilbertSeries

the affine Hilbert-Poincare series

Description

This function computes the affine Hilbert-Poincare series of M.

The grading must be a positive <formula>Z^1</formula>-grading (i.e. <ttref>WeightsMatrix</ttref>

must have a single row with positive entries), and the ordering must be degree compatible (i.e. for a buggy behaviour of cocoa-4, <ttref>Ord</ttref> must have all 1's in the first row). In the standard case, i.e. the weights of all indeterminates are 1, the result is simplified so that the power appearing in the denominator is the dimension of M + 1.

It is exacly the same as <ttref>AffPoincare</ttref>.

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 further details on affine Hilbert functions see the book: Kreuzer, Robbiano Computer Commutative Algebra II, Section 5.6.

Example

  Use R ::= Q[x,y,z];
  AffPoincare(R/Ideal(z^4-1, xz^4-y-3));
(1 + x + x^2 + x^3) / (1-x)^2
-------------------------------

Syntax

HilbertSeries(M:RING or TAGGED(<quotes>Quotient</quotes>)):TAGGED(<quotes>$hp.PSeries</quotes>)

AffHilbert

HilbertSeries

   <type>groebner</type>
   <type>groebner-basic</type>
   <type>hilbert</type>
   <type>quotient</type>
   <type>ring</type>