Hilbert polynomial

Definition

Let $R$ be a graded algebra over a field that occurs as a quotient of a multivariate polynomial ring over a field (finitely many variables) by a graded ideal. Let $M$ be a finitely generated, graded $R$-module. The Hilbert polynomial of $M$, denoted $h_M$ is a polynomial that takes integers to integers, and such that there exists an integer $d_0$ such that for $d\ge d_0$, we have:

$h_M(d)=dim(M_d)$

In other words, the Hilbert polynomial is a polynomial that agrees with the Hilbert function for sufficiently large values of the variable.