C.2 Hilbert function
====================
Let M $=\bigoplus_i M_i$ be a graded module over $K[x_1,..,x_n]$ with 
respect to weights $(w_1,..w_n)$.
The {\bf Hilbert function} of $M$, $H_M$, is defined (on the integers) by
$$H_M(k) :=dim_K M_k.$$
The {\bf Hilbert-Poincare series}  of $M$ is the power series
$$\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty
H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.$$
It turns out that $\hbox{HP}_M(t)$ can be written in two useful ways
for weights $(1,..,1)$:
$$\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over (1-t)^{dim(M)}}$$
where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$.
$Q(t)$ is called the {\bf first Hilbert series},
and $P(t)$ the {\bf second Hilbert series}.
If \hbox{$P(t)=\sum_{k=0}^N a_k t^k$}, and \hbox{$d = dim(M)$},
then \hbox{$H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$}
(the {\bf Hilbert polynomial}) for $s \ge N$.




Generalizing these to quasihomogeneous modules we get
$$\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}$$
where $Q(t)$ is a polynomial in ${\bf Z}[t]$.
$Q(t)$ is called the {\bf first (weighted) Hilbert series} of M.

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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; User manual for <a href="http://www.singular.uni-kl.de/"><i>Singular</i></a> version 2-0-4, October 2002,
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