Explicit Hilbert-Kunz functions of 2 x 2 determinantal rings

TL;DR

This paper derives explicit formulas for the Hilbert-Kunz functions of 2x2 determinantal rings, providing new combinatorial identities and insights into their algebraic structure.

Contribution

It presents the first closed-form expressions for the Hilbert-Kunz functions of 2x2 determinantal rings, linking algebraic and combinatorial results.

Findings

Closed-form formulas for Hilbert-Kunz functions

New binomial identity proved

Connections between algebraic invariants and compositions

Abstract

Let $k[X] = k[x_{i,j}: i = 1,..., m; j = 1,..., n]$ be the polynomial ring in $m n$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2(X)$ the ideal generated by the $2 \times 2$ minors of the generic $m \times n$ matrix $[x_{i,j}]$. We give a closed formulation for the dimensions of the $k$-vector space $k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q))$ as $q$ varies over all positive integers, i.e., we give a closed form for the generalized Hilbert-Kunz function of the determinantal ring $k[X]/I_{2}[X]$. We also give a closed formulation of dimensions of related quotients of $k[X]/I_{2}[X]$. 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.