**Speaker**

Alina Ostafe, UNSW Sidney

**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 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 and Igor Shparlinski.