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

Because $M\mathrm{/}IM$ has finite length over the local ring $(R,\mathfrak{m})$, some power ${\mathfrak{m}}^c$ annihilates it. Equivalently, ${\mathfrak{m}}^{c}M\subseteq IM$. Multiplying by $I^n$ gives ${\mathfrak{m}}^c{I}^{n}M\subseteq I^{n+1}M$ for every $n\ge 0$, so each graded piece $I^{n}M\mathrm{/}{I}^{n+1}M$ is naturally a module over the Artinian local ring $A:=R\mathrm{/}{\mathfrak{m}}^c$.

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: ${\chi }_{I,M}(n)={\sum }_{j=0}^n{\phi }_{I,M}(j)\mathrm{.}$ Summing a polynomial tail again produces a polynomial tail, and summing the standard binomial basis $\left(\genfrac{}{}{0px}{}{j+d-1}{d-1}\right)$ produces $\left(\genfrac{}{}{0px}{}{n+d}{d}\right)$. Therefore ${\chi }_{I,M}(n)$ is eventually a polynomial in binomial form.

Hence the Hilbert-Samuel polynomial exists.