Counting matrices over finite rank multiplicative groups

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).