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This is a Quantitative Study study.

October 1, 2025Open Access

On commuting integer matrices

Key Points

It is shown that the number of commuting integer matrices $orall d imes d$ equates to $N^{10}$ in asymptotic bounds.
An exact expression for $ ext{C}_2(N)$ is derived as $K(2N+1)^5 (1 + o(1))$, with $K>0$ being constant.
Elementary methods were utilized, focusing on upper bounds of divisor correlations to derive results robustly.
The findings confirm a prior speculation by Browning-Sawin-Wang, advancing understanding of matrix commutation.

Abstract

Given d, N N, we define Cd (N) to be the number of pairs of d d matrices A, B with entries in -N, N Z such that AB = BA. We prove that N^10 C₃ (N) N^10, thus confirming a speculation of Browning-Sawin-Wang. We further establish that C₂ (N) = K (2N+1) ⁵ (1 + o (1) ), where K>0 is an explicit constant. Our methods are completely elementary and rely on upper bounds of the correct order for restricted divisor correlations with high uniformity.

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Cite This Study

Chapman et al. (Tue,) studied this question.

synapsesocial.com/papers/68dd91cbfe798ba2fc4986b3 https://doi.org/https://doi.org/10.48550/arxiv.2504.15839