On some matrix counting problems

TL;DR

This paper investigates the frequency of singular and fixed-rank matrices with polynomial entries over integers and finite fields, improving existing bounds on their occurrence.

Contribution

It provides new estimates for the distribution of singular and rank-specific matrices with polynomial entries, extending previous results.

Findings

Improved bounds on the frequency of singular matrices over integers.

Extended analysis to matrices over finite fields modulo prime p.

Enhanced understanding of polynomial-parameterized matrix distributions.

Abstract

We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound of V. Blomer and J. Li (2022).