Build. Blocks
Comb. Appl.
HCA Proving
Arrangements
Branches
Classify
Coding
Deformations
Equidim Part
Existence
Finite Groups
Flatness
Genus
Hilbert Series
Membership
Nonnormal Locus
Normalization
Primdec
Puiseux
Plane Curves
Saturation
Solving
Space Curves
Spectrum
Hilbert Series - An Example
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+2t3 , 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=

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 .

Hilbert series (Mathematical background)


Sao Carlos, 08/02 http://www.singular.uni-kl.de