2020/05/16 by Gilyoung Cheong, Jungin Lee, Cheong, Gilyoung +5
Computer Science · Mathematics · #Advanced Algebra and Geometry #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2005.07846
openalex publication_date 2020/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a positive integer r and a prime power q, we estimate the probability that the characteristic polynomial fA(t) of a random matrix A in GLn(\mathbbFq) is square-free with r (monic) irreducible factors when n is large. We also estimate the analogous probability that fA(t) has r irreducible factors counting with multiplicity. In either case, the main term (log n)r-1((r-1)!n)-1 and the error term O((log n)r-2n-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 \mathbbFq[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 \mathbbFq.