= Solution
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 function>
$$
\chi(M,I;n)=\ell(M/I^nM)
$$
agrees for all sufficiently large $n$ with a polynomial of degree $d$. Its leading term is
$$
\frac{e_I(M)}{d!}n^d,
$$
where $e_I(M)$ is the <Hilbert–Samuel multiplicity>.
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