On some arithmetic statistics for integer matrices

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 a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb{Z}; H)$ with a given characteristic polynomial $f \in\mathbb{Z}[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible. We use our result to address various other questions of arithmetic statistics for matrices from $\mathcal{M}_n(\mathbb{Z}; H)$, eg satisfying certain multiplicative relations. Some of these problems generalise those studied in the scalar case $n=1$ by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices.

Joint works with Kamil Bulinski, Philipp Habegger and Igor Shparlinski.