#### Vol. 275, No. 2, 2015

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Explicit Hilbert–Kunz functions of $2 \times 2$ determinantal rings

### Marcus Robinson and Irena Swanson

Vol. 275 (2015), No. 2, 433–442
##### Abstract

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 $2×2$ minors of the generic $m×n$ matrix $\left[{x}_{i,j}\right]$. We give a closed polynomial formulation for the dimensions of the $k$-vector space $k\left[X\right]∕\left({I}_{2}\left(X\right)+\left({x}_{1,1}^{q},\dots ,{x}_{m,n}^{q}\right)\right)$ 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 $k\left[X\right]∕{I}_{2}\left(X\right)$. We also give a closed formulation of dimensions of other related quotients of $k\left[X\right]∕{I}_{2}\left(X\right)$. 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.

##### Keywords
Hilbert–Kunz function, multiplicity, combinatorial identity
##### Mathematical Subject Classification 2010
Primary: 13D40
Secondary: 05A15, 05A10