Statistics of ranks, determinants and characteristic polynomials of rational matrices
Number Theory
2026-03-04 v3
Abstract
We consider the set of matrices with rational entries having numerator and denominator of size at most and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such matrices with a given determinant, or a given characteristic polynomial.
Keywords
Cite
@article{arxiv.2401.10086,
title = {Statistics of ranks, determinants and characteristic polynomials of rational matrices},
author = {Muhammad Afifurrahman and Vivian Kuperberg and Alina Ostafe and Igor E. Shparlinski},
journal= {arXiv preprint arXiv:2401.10086},
year = {2026}
}
Comments
his is a slight revision of the published version (Forum Math. 2025; v.37, 1259-1289), in which the main and error terms in Theorem 2.6 and Lemma 3.10 all contain an extra factor of $\log H$. The authors are grateful to Nick Rome for noticing this