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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
HM (k) := dimKMk.
The Hilbert-Poincare series of M is the power series
           ∞             ∞
HPM (t) := ∑  HM  (i)ti = ∑  dimKMi  ⋅ti.
          i=− ∞          i=− ∞
It turns out that HPM(t) can be written in two useful ways for weights (1,..,1):
HPM  (t) = -Q-(t)n-= ----P(td)im(M-)
          (1− t)   (1− t)
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+sk1 d1 ) (the Hilbert polynomial) for s N.

Generalizing these to quasihomogeneous modules we get
         ----Q-(t)----
HPM  (t) = Πni=1(1 − twi)
where Q(t) is a polynomial in Z[t]. Q(t) is called the first (weighted) Hilbert series of M.

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