Computing explicit isomorphisms with full matrix algebras 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.

15 pages, contains updated grant numbers

References in corpus (2)

• Explicit n-descent on elliptic curves. III. Algorithms
• Factoring Ore polynomials over ${\mathbb{F}}_q(t)$ is difficult

Cited by in corpus (1)

• Explicit equivalence of quadratic forms over ${\mathbb{F}}_q(t)$