On matrices commuting with their Frobenius

TL;DR

This paper investigates the asymptotic count of matrices over finite fields that commute with their Frobenius automorphism, providing solutions for specific matrix classes and outlining steps for the general case.

Contribution

It offers new results on counting matrices commuting with their Frobenius automorphism over finite fields, including special cases and a framework for the general case.

Findings

Exact counts for 2x2 matrices commuting with Frobenius

Results for diagonalizable matrices over finite fields

Analysis of matrices commuting with entire Frobenius orbits

Abstract

The Frobenius of a matrix $M$ with coefficients in $\bar{\mathbb F}_p$ is the matrix $\sigma(M)$ obtained by raising each coefficient to the $p$-th power. We consider the question of counting matrices with coefficients in $\mathbb F_q$ which commute with their Frobenius, asymptotically when $q$ is a large power of $p$. We give answers for matrices of size $2$, for diagonalizable matrices, and for matrices whose eigenspaces are defined over $\mathbb F_p$. Moreover, we explain what is needed to solve the case of general matrices. We also solve (for both diagonalizable and general matrices) the corresponding problem when one counts matrices $M$ commuting with all the matrices $\sigma(M)$, $\sigma^2(M)$, $\ldots$ in their Frobenius orbit.