The conjugacy problem for two by two matrices over polynomial rings

TL;DR

This paper presents an effective method to determine whether two 2x2 matrices over a polynomial ring in one variable over a finite field are conjugate, solving a classical algebraic problem.

Contribution

It provides the first effective solution to the conjugacy problem for 2x2 matrices over polynomial rings over finite fields.

Findings

Developed an algorithm for the conjugacy problem

Proved the problem is decidable in this setting

Applied the method to finite field polynomial rings

Abstract

We give an effective solution of the conjugacy problem for two by two matrices over the polynomial ring in one variable over a finite field.