Pairs of commuting integer matrices

TL;DR

This paper establishes bounds on the number of integer matrix pairs that commute, using Fourier analysis and finite field exponential sums, advancing understanding of matrix commutation properties.

Contribution

It introduces new bounds on commuting integer matrices and combines Fourier analysis with finite field exponential sum techniques.

Findings

Derived upper and lower bounds for commuting matrix pairs

Applied Fourier analysis to matrix counting problems

Connected matrix commutation to finite field exponential sums

Abstract

We prove upper and lower bounds on the number of pairs of commuting $n\times n$ matrices with integer entries in $[-T,T]$, as $T\to \infty$. Our work uses Fourier analysis and leads us to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket.