Counting matrices over finite rank multiplicative groups

TL;DR

This work analyzes counting problems for $m\times n$ matrices with entries from a finite subset ${\mathcal{A}}$ of a rank-$\varrho$ multiplicative subgroup ${\Gamma}$ in a characteristic-zero field, focusing on rank, determinant, and characteristic polynomial constraints. It develops uniform upper bounds in terms of $A=\#\mathcal{A}$ by combining Subspace-Theorem techniques (via Amoroso–Viada) with refined counts of linear equations in finite rank multiplicative groups, and reduces matrix-counting questions to structured linear relations. The main results provide explicit bounds: for rank constraints, $\#\mathcal{R}_{m,n}({\mathcal{A}}; r)$ with a break at $2m\le n+r$, determinant bounds improving trivial ones and tightness statements, and polynomial-bound bounds $\#\mathcal{P}_n({\mathcal{A}}; f) \ll A^{\alpha(n)}$ with $\alpha(n)/n^{2} \to 3/4$; these are complemented by detailed counts of non-degenerate and arbitrary solutions to linear equations in $\Gamma$. The findings advance the arithmetic-statistical understanding of matrices over structured multiplicative sets and suggest further directions, including symmetry constraints, commuting matrices, and extensions to positive characteristic settings.

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).

Theorems & Definitions (17)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4