## Abstract

Given a positive integer r and a prime power q, we estimate the probability that the characteristic polynomial f_{A}(t) of a random matrix A in GL_{n}(F_{q}) is square-free with r (monic) irreducible factors when n is large. We also estimate the analogous probability that f_{A}(t) has r irreducible factors counting with multiplicity. In either case, the main term (logn)^{r−1}((r−1)!n)^{−1} and the error term O((logn)^{r−2}n^{−1}), whose implied constant only depends on r but not on q nor n, coincide with the probability that a random permutation on n letters is a product of r disjoint cycles. The main ingredient of our proof is a recursion argument due to S. D. Cohen, which was previously used to estimate the probability that a random degree n monic polynomial in F_{q}[t] is square-free with r irreducible factors and the analogous probability that the polynomial has r irreducible factors counting with multiplicity. We obtain our result by carefully modifying Cohen's recursion argument in the matrix setting, using Reiner's theorem that counts the number of n×n matrices with a fixed characteristic polynomial over F_{q}.

Original language | English |
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Pages (from-to) | 100-128 |

Number of pages | 29 |

Journal | Linear Algebra and Its Applications |

Volume | 655 |

DOIs | |

Publication status | Published - 2022 Dec 15 |

### Bibliographical note

Funding Information:We thank Yifeng Huang, Nathan Kaplan, Ofir Gorodetsky, and Michael Zieve for helpful conversations. G. Cheong was supported by NSF grant DMS-1162181 and the Korea Institute for Advanced Study for his visits to the institution regarding this research. We deeply appreciate the referee for quick and detailed comments on the previous draft of this paper. J. Lee was supported by a KIAS Individual Grant ( MG079602 ) at Korea Institute for Advanced Study. H. Nam was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1F1A106231911 ). M. Yu was supported by a KIAS Individual Grant ( SP075201 ) via the Center for Mathematical Challenges at Korea Institute for Advanced Study. He was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01007604 ). This research was also supported by the Yonsei University Research Fund of 2022-22-0125 .

Publisher Copyright:

© 2022 Elsevier Inc.

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics