The Hilbert-Samuel function agrees eventually with a rational polynomial in binomial form

Every simple module over the local ring $R$ is $R\mathrm{/}\mathfrak{m}$: a nonzero element generates a simple module, whose annihilator is a maximal ideal, necessarily $\mathfrak{m}$. Along a composition series of the finite-length module $M\mathrm{/}IM$, each multiplication by $\mathfrak{m}$ moves into the preceding term. Thus ${\mathfrak{m}}^c(M\mathrm{/}IM)=0$ for some $c\ge 1$, or ${\mathfrak{m}}^{c}M\subseteq IM$. Multiplying by $I^n$ gives ${\mathfrak{m}}^c{I}^{n}M\subseteq I^{n+1}M$ for every $n\ge 0$. The quotient $A:=R\mathrm{/}{\mathfrak{m}}^c$ is Artinian: $\mathfrak{m}$ is finitely generated by [L3], so each factor ${\mathfrak{m}}^i\mathrm{/}{\mathfrak{m}}^{i+1}$ for $0\le i<c$ is a finite-dimensional vector space over $R\mathrm{/}\mathfrak{m}$; their finite filtration gives $A$ finite length as an $R$-module. Each graded piece $I^{n}M\mathrm{/}{I}^{n+1}M$ is naturally an $A$-module.

By [L3], choose generators $x_1,\dots ,x_r$ of $I$, and choose generators $m_1,\dots ,m_s$ of $M$. The classes of the $m_j$ in degree $0$ generate ${\mathrm{gr}}_I(M)$ over the standard graded $A$-algebra $A[T_1,\dots ,T_r]$, where $T_i$ acts by multiplication with the class of $x_i$. Indeed every class in $I^{n}M\mathrm{/}{I}^{n+1}M$ is represented by a finite sum of monomials $x_{i_1}\cdots x_{i_n}{m}_j$.

Length over $R$ and over $A=R\mathrm{/}{\mathfrak{m}}^c$ agree on each module $I^{n}M\mathrm{/}{I}^{n+1}M$, because that quotient is annihilated by ${\mathfrak{m}}^c$ and has the same submodules in either category. Applying [L2] to the finite graded $A[T_1,\dots ,T_r]$-module ${\mathrm{gr}}_I(M)$, the function $n⟼{\mathrm{\ell }}_R(I^{n}M\mathrm{/}{I}^{n+1}M)$ agrees for large $n$ with a polynomial $Q(n)$. Equivalently, the Hilbert series of ${\mathrm{gr}}_I(M)$ is rational with denominator a power of $(1-t)$.

By [L1], the Hilbert-Samuel function is the cumulative sum of these graded-piece lengths: ${\chi }_{I,M}(n)={\sum }_{j=0}^n{\phi }_{I,M}(j)\mathrm{.}$ The rational Hilbert series in [L2] has an integral Laurent numerator and denominator $(1-t{)}^r$; cumulative summation divides it by $(1-t)$. Its large-degree coefficients are therefore integer linear combinations of shifted binomial polynomials $\left(\genfrac{}{}{0px}{}{n-k+r}{r}\right)$. The finite-difference identity for integer-valued polynomials expresses each such shift as an integer linear combination of $\left(\genfrac{}{}{0px}{}{n+j}{j}\right)$, $0\le j\le r$. Thus ${\chi }_{I,M}(n)$ is eventually a polynomial in the claimed binomial form.

Hence the Hilbert-Samuel polynomial exists.