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 $d \geq d_{0}$ , we have:

$h_{M} (d) = d i m (M_{d})$

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