## C.2 Hilbert function

Let M be a graded module over 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 . 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 . is called the first (weighted) Hilbert series of M.

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