Pairs of commuting integer matrices

TL;DR

The paper bounds the number of commuting integer matrix pairs of size $n$ with entries in $[-T,T]$ by analyzing the commuting variety and employing harmonic analysis. Central to the approach is a flatness result for the commutator map on the open locus $\mathsf{V}_n\setminus\{0\}$, established via an explicit fibre-dimension bound over finite fields and spread to characteristic zero. Exponential sums over finite fields are controlled using a stratification of Fouvry–Katz combined with a new flatness refinement, and the final bound $N(T) \ll_n T^{n^2+2-\frac{2}{n+1}}$ follows from a careful Poisson-summation/diophantine-geometry analysis and optimization over the auxiliary prime $p$. The work connects arithmetic geometry (Lang-Weil-type counts, fibre dimensions) with Fourier-analytic methods to constrain lattice points on the commuting variety, achieving improvements over prior bounds for large $n$.

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.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Theorem 2.2
• proof : Proof of the equivalence of Theorems \ref{['THM:flat']} and \ref{['THM:dimension']}
• Lemma 3.1
• proof
• Theorem 4.1
• Lemma 5.1