# Uniform Probability and Natural Density of Mutually Left Coprime Polynomial Matrices over Finite Fields

## TL;DR

This paper compares uniform probability and natural density in the context of mutually left coprime polynomial matrices over finite fields, showing asymptotic agreement but differences in exact values.

## Contribution

It provides explicit calculations of probabilities for coprimality of polynomial matrices and compares different probability measures, extending scalar results to matrix cases.

## Key findings

- Asymptotic coincidence of probabilities under different measures
- Exact probability formulas differ but converge asymptotically
- Natural density estimates for polynomial matrices

## Abstract

We compute the uniform probability that finitely many polynomials over a finite field are pairwise coprime and compare the result with the formula one gets using the natural density as probability measure. It will turn out that the formulas for the two considered probability measures asymptotically coincide but differ in the exact values. Moreover, we calculate the natural density of mutually left coprime polynomial matrices and compare the result with the formula one gets using the uniform probability distribution. The achieved estimations are not as precise as in the scalar case but again we can show asymptotic coincidence.