# Unimodular Polynomial Matrices over Finite Fields

## TL;DR

This paper investigates the properties and enumeration of unimodular polynomial matrices over finite fields, providing new proofs and resolving conjectures related to their probability and structure.

## Contribution

It offers a novel proof of existing results using control theory and resolves open questions about the probability of unimodularity in matrix polynomials.

## Key findings

- Confirmed the probability that a matrix polynomial is unimodular
- Provided a new proof of a theorem on splitting subspaces
- Resolved a conjecture on unimodular polynomial matrices

## Abstract

We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite field. As an application of our results we give a new proof of a theorem of Chen and Tseng which answers a question of Niederreiter on splitting subspaces. We use our results to affirmatively resolve a conjecture on the probability that a matrix polynomial is unimodular.