Pairs of commuting integer matrices
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915618944450560 |
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| author | Browning, Tim Sawin, Will Wang, Victor Y. |
| author_facet | Browning, Tim Sawin, Will Wang, Victor Y. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pairs of commuting integer matrices Browning, Tim Sawin, Will Wang, Victor Y. Number Theory 11D45 (11T23, 15B33, 15B36) 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. |
| title | Pairs of commuting integer matrices |
| topic | Number Theory 11D45 (11T23, 15B33, 15B36) |
| url | https://arxiv.org/abs/2409.01920 |