Counting zero kernel pairs over a finite field

TL;DR

This paper provides a new elementary proof for counting reachable matrix pairs over finite fields by analyzing zero kernel pairs, connecting to unimodular matrices and proposing a density conjecture.

Contribution

It introduces a simplified proof for the enumeration of zero kernel pairs and links this problem to unimodular matrix densities over polynomial rings.

Findings

Established a new proof for the count of zero kernel pairs

Connected enumeration problem to unimodular matrix density results

Proposed a conjecture on the density of unimodular matrix polynomials

Abstract

Helmke et al. have recently given a formula for the number of reachable pairs of matrices over a finite field. We give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field. We show that the problem is equivalent to certain other enumeration problems and outline a connection with some recent results of Guo and Yang on the natural density of rectangular unimodular matrices over $\Fq[x]$. We also propose a new conjecture on the density of unimodular matrix polynomials.