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C.2 Hilbert function
Let M = ⊕
iMi be a graded module over K[x1,..,xn]
with respect to weights (w1,..wn). The Hilbert function of M, HM, is defined (on the integers)
by
The
Hilbert-Poincare series of M is the power series
It
turns out that HPM(t) can be written in two useful ways for weights (1,..,1):
where
Q(t) and P(t) are polynomials in Z[t]. Q(t) is called the first Hilbert series, and P(t) the second Hilbert
series. If P(t) = ∑
k=0Naktk, and d = dim(M), then HM(s) = ∑
k=0Nak (d+s−k−1
d−1 )
(the Hilbert polynomial)
for s ≥ N.
Generalizing these to quasihomogeneous modules we get
where Q(t) is a polynomial in Z[t]. Q(t) is called the first (weighted) Hilbert series of M.
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