On the Frobenius complexity of determinantal rings

TL;DR

This paper calculates the Frobenius complexity of determinantal rings formed by 2x2 minors over fields of prime characteristic, revealing it approaches m-1 as p increases.

Contribution

It provides an explicit computation of Frobenius complexity for determinantal rings and analyzes its asymptotic behavior as the characteristic grows.

Findings

Frobenius complexity is computed explicitly for the given rings.

As p approaches infinity, the Frobenius complexity tends to m-1.

The study enhances understanding of Frobenius invariants in algebraic geometry.

Abstract

We compute the Frobenius complexity for the determinantal ring of prime characteristic $p$ obtained by modding out the $2 \times 2$ minors of an $m \times n$ matrix of indeterminates, where $m > n \ge 2$. We also show that, as $p \to \infty$, the Frobenius complexity approaches $m-1$.