Hilbert series --<b> SINGULAR</b> code

Compute the (first and second) Hilbert series of a homogeneous ideal.

Twisted cubic in P³ : V = ( xz-y²=0 , xw-yz=0 , yw-z²=0 )
ring r=0,(x,y,z,w),dp; ideal i=xz-y2,xw-yz,yw-z2; i=std(i); hilb(i); ==> 1 t^0 ==> -3 t^2 ==> 2 t^3 ==> ==> 1 t^0 ==> 2 t^1 ==> codimension = 2 dimension = 2 degree = 3

1st Hilbert series: Q(t) = 1-3t²+2t² , 2nd Hilbert series: P(t) = 1+2t .

This result can be used to compute the Hilbert polynomial H of M = K[x,y,z,w] / I :

$\displaystyle P(t) = \sum_{j=0}^N a_j\cdot t^j \;\; \Longrightarrow \;\; H(t)=\sum_{j=0}^N a_j \cdot\binom{\dim(I)+t-j-1}{\dim(I)-1}$

intvec a=hilb(i,2); ring s=0,t,ls; poly h; int j; for (j=1; j<=size(a); j=j+1){h=h+a[j]*(t-j+2);} h; ==> 1+3t
Hilbert polynomial: H(t) = 1+3t .

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