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.