Pairs of commuting integer matrices
Tim Browning, Will Sawin, Victor Y. Wang

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 matrices with integer entries in , as . 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
Topicsgraph theory and CDMA systems · Advanced Optimization Algorithms Research · Matrix Theory and Algorithms
