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Counting matrices
over finite rank multiplicative groups
Aaron Manning, Alina Ostafe and Igor E. Shparlinski |
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Vol. 340 (2026), No. 1, 115–138
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Abstract |
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Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets of finite rank multiplicative groups in fields of characteristic zero. We obtain upper bounds, in terms of the size of , on the number of such matrices of a given rank, with a given determinant and with a prescribed characteristic polynomial. In particular, in the case of ranks, our results can be viewed as a statistical version of work by Alon and Solymosi (2023). |
Keywords
matrices over finite rank multiplicative groups, rank,
determinant, characteristic polynomial
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Mathematical Subject Classification
Primary: 11C20, 15B36, 60B20
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Milestones
Received: 19 February 2025
Revised: 25 August 2025
Accepted: 18 September 2025
Published: 31 October 2025
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Authors |
| Aaron Manning | |
| School of Mathematics and
Statistics University of New South Wales Sydney NSW Australia |
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| Alina Ostafe | |
| School of Mathematics and
Statistics University of New South Wales Sydney NSW Australia |
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| Igor E. Shparlinski | |
| School of Mathematics and
Statistics University of New South Wales Sydney NSW Australia |
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| © 2026 MSP (Mathematical Sciences Publishers). |
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