Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices

TL;DR

This paper establishes new bounds on the number of integer matrices with a given characteristic polynomial, and explores multiplicative relations among matrices, extending scalar case results to matrices.

Contribution

It provides uniform upper bounds on matrices with fixed characteristic polynomial and investigates multiplicative dependence in matrix tuples, generalizing scalar results.

Findings

New upper bounds on matrices with fixed characteristic polynomial.

Bounds on the number of matrix tuples satisfying multiplicative relations.

Computed the degree of the variety of matrices with a fixed characteristic polynomial.

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. Using this result, among others, we obtain upper and lower bounds on the number of $s$-tuples of matrices from $\mathcal{M}_n(\mathbb Z; H)$, satisfying various multiplicative relations, including multiplicative dependence and bounded generation of a subgroup of $\mathrm{GL}_n(\mathbb Q)$. 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. Motivated by these problems, we also prove various properties of the variety of complex matrices with fixed characteristic polynomial, including computing the degree of this variety.