The s-multiplicity function of 2x2-determinantal rings

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 Grobner 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 $s$.