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Definition 2..8
Let M be a graded module over
.
The
Hilbert series of M is the power series
.
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)/(1t)^{n}, where Q(t) is a polynomial in t and
n ist the number of variables in
.
 2.

H(M)(t)=P(t)/(1t)^{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=c3bd2,bcad,b3a2c,ac2b2d;
// the output of hilb is Q, then P:
hilb(J);
Next: 2.4 Applications
Up: 2.3 Basic properties
Previous: 2.3.2 Elimination
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