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C.2 Hilbert function
Let M
639#639 be a graded module over 640#640 with
respect to weights 641#641.
The Hilbert function of 13#13, 642#642, is defined (on the integers) by
643#643
The Hilbert-Poincare series of 13#13 is the power series
644#644
It turns out that
645#645 can be written in two useful ways
for weights 646#646:
647#647
where 648#648 and 649#649 are polynomials in 650#650.
648#648 is called the first Hilbert series,
and 649#649 the second Hilbert series.
If
651#651, and 652#652,
then
653#653
654#654
(the Hilbert polynomial) for 655#655.
Generalizing this to quasihomogeneous modules we get
656#656
where 648#648 is a polynomial in 650#650.
648#648 is called the first (weighted) Hilbert series of M.
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