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For an ideal of definition $I$ of a Noetherian local ring and a finitely generated module $M$, the Hilbert–Samuel function isFor sufficiently large $n$, it equals the Hilbert-Samuel polynomial.

An ideal of definition of a Noetherian local ring $(A,\mathfrak{m})$ is an $\mathfrak{m}$-primary ideal $I$; equivalently, $\sqrt{I}=\mathfrak{m}$, or some power ${\mathfrak{m}}^r$ is contained in $I$.

If $M$ is a finitely generated module of dimension $d$, the Hilbert–Samuel functionagrees for all sufficiently large $n$ with a polynomial of degree $d$. Its leading term iswhere $e_I(M)$ is the Hilbert–Samuel multiplicity.