Computing explicit isomorphisms with full matrix algebras over $\mathbb{F}_q(x)$

TL;DR

This paper presents a polynomial-time deterministic algorithm for explicitly computing isomorphisms between finite-dimensional algebras over $\\mathbb{F}_q(x)$ and full matrix algebras, leveraging order intersection techniques.

Contribution

It introduces a novel polynomial-time algorithm that uses an oracle for polynomial factorization to find explicit isomorphisms of algebras over rational function fields.

Findings

Algorithm operates in polynomial time with respect to input size.

Uses order intersection to compute algebra isomorphisms.

Applicable to algebras given by structure constants over $\\mathbb{F}_q(x)$.

Abstract

We propose a polynomial time $f$-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$) for computing an isomorphism (if there is any) of a finite dimensional $\mathbb{F}_q(x)$-algebra $A$ given by structure constants with the algebra of $n$ by $n$ matrices with entries from $\mathbb{F}_q(x)$. The method is based on computing a finite $\mathbb{F}_q$-subalgebra of $A$ which is the intersection of a maximal $\mathbb{F}_q[x]$-order and a maximal $R$-order, where $R$ is the subring of $\mathbb{F}_q(x)$ consisting of fractions of polynomials with denominator having degree not less than that of the numerator.