Statistics of ranks, determinants and characteristic polynomials of rational matrices

Abstract

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ 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.

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