Is there a commonly used name and notation for $\beta(n)=\dim_{A/\mathbf{m}}\mathbf{m}^{n}/\mathbf{m}^{n+1}$, where $(A,\mathbf{m})$ is a local ring?

Question

I've recently found myself doing some work on local rings, and I found the following quantity keeps popping up-

Let $A$ be a local commutative unital ring, with maximal ideal $\newcommand{\mfr}{\mathbf} \mfr{m}$. For any $n\in\mathbb{N}$, put $\beta(n)=\dim_{A/\mfr{m}}\mfr{m}^n/\mfr{m}^{n+1}$.

It seems quite plausible that one might consider this quantity an important invariant of the ring $A$. I was only wondering if this quantity has a commonly used name, and possibly notation.

Thank you, shai

Answer 1

Per HeinrichD's comment, I am rewriting my comment as an answer.

The Hilbert-Samuel function is $\alpha(n) = \text{length}(A/\mathbf{m}^n)$. The function $\beta(n)$ is $\alpha(n)-\alpha(n-1)$, the first difference of the Hilbert-Samuel function.

If $A$ is Noetherian, then there exists a unique numerical polynomial $p(x)\in \mathbb{Q}[x]$ such that for all $n\geq n_0$, $\alpha(n)$ equals $P(n)$. The polynomial $p(x)$ is the Hilbert-Samuel polynomial of $A$ with respect to the $\mathbf{m}$-adic filtration. The degree $d$ of $p(x)$ equals the Krull dimension of $A$. The leading term of $p(x)$ is $ex^d/d!$, where the integer $e$ is the Hilbert-Samuel multiplicity of $A$ with respect to $\mathbf{m}$.