Pairs of commuting integer matrices

Tim Browning; Will Sawin; Victor Y. Wang

doi:10.1007/s00208-025-03285-5

Pairs of commuting integer matrices

Tim Browning, Will Sawin, Victor Y. Wang

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove upper and lower bounds on the number of pairs of commuting n×n matrices with integer entries in [-T,T], as T→∞. Our work uses Fourier analysis and leads 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.

Original language	English (US)
Pages (from-to)	1863-1880
Number of pages	18
Journal	Mathematische Annalen
Volume	393
Issue number	2
DOIs	https://doi.org/10.1007/s00208-025-03285-5
State	Published - Oct 2025

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00208-025-03285-5

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AB - We prove upper and lower bounds on the number of pairs of commuting n×n matrices with integer entries in [-T,T], as T→∞. Our work uses Fourier analysis and leads 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.

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Pairs of commuting integer matrices

Abstract

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