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



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?
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 $\mathbf{m}$.


