Let
$k\left[X\right]=k\left[{x}_{i,j}:i=1,\dots ,m;j=1,\dots ,n\right]$ be the
polynomial ring in
$mn$
variables
${x}_{i,j}$ over a field
$k$ of arbitrary characteristic.
Denote by
${I}_{2}\left(X\right)$ the ideal
generated by the
$2\times 2$
minors of the generic
$m\times n$
matrix
$\left[{x}_{i,j}\right]$.
We give a closed polynomial formulation for the dimensions of the
$k$vector
space
$k\left[X\right]\u2215\left({I}_{2}\left(X\right)+\left({x}_{1,1}^{q},\dots ,{x}_{m,n}^{q}\right)\right)$
as
$q$
varies over all positive integers, i.e., we give a closed polynomial form
for the generalized Hilbert–Kunz function of the determinantal ring
$k\left[X\right]\u2215{I}_{2}\left(X\right)$. We
also give a closed formulation of dimensions of other related quotients of
$k\left[X\right]\u2215{I}_{2}\left(X\right)$. In
the process we establish a formula for the numbers of some compositions (ordered
partitions of integers), and we give a proof of a new binomial identity.
