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Statistics of ranks, determinants and characteristic polynomials of rational matrices

  • Muhammad Afifurrahman , Vivian Kuperberg , Alina Ostafe and Igor E. Shparlinski EMAIL logo
Published/Copyright: October 2, 2024

Abstract

We consider the set of m × n matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions.

MSC 2020: 11C20; 15B36; 15B52

Communicated by Chantal David


Award Identifier / Grant number: DP200100355

Award Identifier / Grant number: DP230100530

Award Identifier / Grant number: DMS-2202128

Funding statement: During the preparation of this paper, Muhammad Afifurrahman, Alina Ostafe and Igor Shparlinski were supported by Australian Research Council Grants DP200100355 and DP230100530, Muhammad Afifurrahman was also supported by a UNSW Tuition Fee Scholarship, and Vivian Kuperberg was supported by NSF Mathematical Sciences Research program grant DMS-2202128.

Acknowledgements

The authors are grateful to the referee for carefully reading our paper and for useful comments. V. Kuperberg, A. Ostafe, I. Shparlinski would also like to thank the Mittag-Leffler Institute for its hospitality and excellent working environment.

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Received: 2024-03-04
Revised: 2024-07-29
Published Online: 2024-10-02
Published in Print: 2025-06-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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