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C.2 Hilbert function
Let M
641#641 be a graded module over 642#642 with
respect to weights 643#643.
The Hilbert function of 13#13, 644#644, is defined (on the integers) by
645#645
The Hilbert-Poincare series of 13#13 is the power series
646#646
It turns out that
647#647 can be written in two useful ways
for weights 648#648:
649#649
where 650#650 and 651#651 are polynomials in 652#652.
650#650 is called the first Hilbert series,
and 651#651 the second Hilbert series.
If
653#653, and 654#654,
then
655#655
656#656
(the Hilbert polynomial) for 657#657.
Generalizing this to quasihomogeneous modules we get
658#658
where 650#650 is a polynomial in 652#652.
650#650 is called the first (weighted) Hilbert series of M.
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