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 : 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)