The s-multiplicity function of 2x2-determinantal rings

TL;DR

This paper extends the concept of s-multiplicity to determinantal rings, providing a closed-form formula for certain lengths using Gröbner bases, and demonstrating that these lengths are eventually polynomial in q.

Contribution

It introduces a new formula for s-multiplicity in determinantal rings and proves the length function is eventually polynomial in q.

Findings

Length function is eventually polynomial in q.

Provides a closed-form expression using Gröbner bases.

Generalizes previous work on s-multiplicity.

Abstract

This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m \times n$-matrix of variables, we utilize Gr\"obner bases to give a closed form the length $\lambda( k[X] / (I_2(X) + \mathfrak{m}^{ \lceil sq \rceil} + \mathfrak{m}^{[q]} ))$ where $s \in \mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {\it polynomial} function of $q$ for all $s$.