On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant

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 bounds on the number of matrices from ${\mathcal{M}}_n(\mathbb{Z};H)$, for which the characteristic polynomial has a fixed discriminant $d$. When $d=0$, this corresponds to counting matrices with a repeated eigenvalue, and thus is related to counting non-diagonalisable matrices. For $d\ne 0$, this problem seems not to have been studied previously, while for $d=0$, both our approach and the final result improve on those of A. J. Hetzel, J. S. Liew and K. Morrison (2007).