Number of characteristic polynomials of matrices with bounded height

Abstract

We consider the set ${\mathcal{M}}_n(\mathbb{Z};H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of distinct irreducible characteristic polynomials which correspond to these matrices and thus on the number of distinct eigenvalues of these matrices. In particular, we improve some results of A.~Abrams, Z.~Landau, J.~Pommersheim and N.~Srivastava (2022).