Hilbert series

$\displaystyle HP_M(t) = \sum_{i=-\infty}^\infty H_M(i)\cdot t^i = \sum_{i=-\infty}^\infty \dim_K M_i\cdot t^i$

It turns out that HP_M(t) can be written in two useful ways for weights (1,...,1) :

$$\displaystyle HP_M(t) = \frac{Q(t)}{(1-t)^n}=\frac{P(t)}{(1-t)^{\dim(M)}} </DIV> <BR CLEAR=$
where Q(t) and P(t) are polynomials with coefficients in Z.

Q(t) is called the first Hilbert series,
P(t) is called the second Hilbert series.

For general w₁,...,w_n :

$\displaystyle HP_M(t) = \frac{Q(t)}{\prod\limits_{i=1}^n (1-t^{w_i})}$

where Q(t) is a polynomial in Z[t], the first (weighted) Hilbert series of M . An Example