2.3.3 Hilbert series

Definition 2..8   Let M be a graded module over $K[\underline{x}]$. The Hilbert series of M is the power series

$$H(M)(t)=\sum_{t=-\infty}^\infty dim_K M_i t^i$$

.

Lemma 2..9   Let < be a (positive or negative) degree ordering and H(M) the Hilbert function of (the homogenization of) I. Then H(M)=H(L(M)).

Remark 2..10   It turns out that H(M)(t) can be written in two usefule ways:

1.

H(M)(t)=Q(t)/(1-t)ⁿ, where Q(t) is a polynomial in t and n ist the number of variables in $K[\underline{x}]$.

2.

H(M)(t)=P(t)/(1-t)^{dim M} where P(t) is a polynomial and deg M=P(1).

3.

vector space dimension dim_K(M)=dim_K(L(M)).

Remark 2..11   Let < be a degree ordering.

• Krull dimension: dim(M)=dim(L(M)).
• degree (for a positive degree ordering) resp. multiplicity (for a negative degree ordering) is equal for M and L(M).

SINGULAR example:

// the rational quartic curve J in P^3:
ring R=0,(a,b,c,d),dp;
ideal J=c3-bd2,bc-ad,b3-a2c,ac2-b2d;
// the output of hilb is Q, then P:
hilb(J);