Counting matrices over finite rank multiplicative groups

TL;DR

This paper investigates the number of matrices over finite rank multiplicative groups with entries from finite subsets, providing bounds related to rank, determinant, and characteristic polynomial, extending prior combinatorial results.

Contribution

It introduces new upper bounds on the count of matrices with specified properties over finite rank multiplicative groups, generalizing previous combinatorial and algebraic results.

Findings

Derived bounds on the number of matrices with fixed rank and determinant

Extended statistical analysis of matrices over finite rank groups

Connected results to prior work by Alon and Solymosi

Abstract

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We obtain upper bounds, in terms of the size of $\mathcal A$, on the number of such matrices of a given rank, with a given determinant and with a prescribed characteristic polynomial. In particular, in the case of ranks, our results can be viewed as a statistical version of work by Alon and Solymosi (2003).