Hilbert function

Definition

Let $R$ be a Noetherian local ring with maximal ideal $M$ and residue field $k=R/M$. The Hilbert function of $R$ is defined as:

$H(n)=di{m}_k(M^n/M^{n+1})$

For large $n$, $H(n)$ agrees with a polynomial, called the Hilbert polynomial of $R$.