|  |  C.2 Hilbert function Let M be a graded module over ![$K[x_1,..,x_n]$](sing_603.png) with
respect to weights  .
The Hilbert function of  ,  , is defined (on the integers) by 
 The Hilbert-Poincare series  of
  is the power series 
 It turns out that
  can be written in two useful ways
for weights  : 
 where
  and  are polynomials in ![${\bf Z}[t]$](sing_613.png) .  is called the first Hilbert series,
and  the second Hilbert series.
If  , and  ,
then    (the Hilbert polynomial) for  . 
 Generalizing this to quasihomogeneous modules we get
 
 where
  is a polynomial in ![${\bf Z}[t]$](sing_613.png) .  is called the first (weighted) Hilbert series of M. 
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