Hilbert-Samuel functions
yihuang · Fri Aug 29, 2008 12:05 am
Hi! Let (R,m) be a Noetherian local ring. I just wonder if there is any SINGULAR codes that can compute the Poincare series of the Hilbert-Samuel functions for m-primary ideals. Thanks.
Hilbert-Samuel functions
Gerhard Pfister · Mon Mar 30, 2009 9:32 pm
Singular has a command which computes the numerator Q(t) for the Hilbert-Poincare series (the denominator is (1-t)^n,
n the number of variables):
The latter output has to be interpreted as follows:
if v= (v0, ... , vd, 0) then Q(t) = sum_{i=0}^{d} vi t^i.
More details can be found in the book
G.-M. Greuel / G. Pfister http://www.mathematik.uni-kl.de/%7Epfister/Artikel/buchGMG.ps.gz: A Singular Introduction to Commutative Algebra, Springer 2002 second edition 2007 (pages 315 ff).
n the number of variables):
Code:
ring A=0,(t,x,y,z),ds;
ideal I=x5y2,x3,y3,xy4,xy7;
intvec v = hilb(std(I),1);
v;
//-> 1,0,0,-2,0,0,1,0
ideal I=x5y2,x3,y3,xy4,xy7;
intvec v = hilb(std(I),1);
v;
//-> 1,0,0,-2,0,0,1,0
The latter output has to be interpreted as follows:
if v= (v0, ... , vd, 0) then Q(t) = sum_{i=0}^{d} vi t^i.
More details can be found in the book
G.-M. Greuel / G. Pfister http://www.mathematik.uni-kl.de/%7Epfister/Artikel/buchGMG.ps.gz: A Singular Introduction to Commutative Algebra, Springer 2002 second edition 2007 (pages 315 ff).