Hilbert–Samuel function

Summary

Let $M$ be a finitely generated module over a Noetherian local ring, and let $I$ be an ideal primary to the maximal ideal. The Hilbert–Samuel function of $M$ is $n \mapsto \ell(M/I^n M)$, where $\ell$ is the length. For large $n$ it agrees with a polynomial, the Hilbert–Samuel polynomial. If $M \neq 0$, the degree of this polynomial is the dimension of $M$, and its leading coefficient gives the multiplicity.