On Efficient Noncommutative Polynomial Factorization via Higman Linearization

TL;DR

This paper introduces a randomized polynomial-time algorithm for factorizing noncommutative polynomials over finite fields by transforming them into linear matrices using Higman's linearization, enabling efficient factorization.

Contribution

It presents the first efficient randomized algorithm for noncommutative polynomial factorization leveraging Higman's linearization and algebraic branching programs.

Findings

Algorithm runs in polynomial time in size, variables, and log of the field size.

Successfully factorizes noncommutative polynomials into irreducible factors.

Utilizes Higman's linearization and matrix invariant subspace computation techniques.

Abstract

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F of polynomials in noncommuting variables x1,x2,...,xn over the field F. We obtain the following result Given a noncommutative algebraic branching program of size s computing a noncommutative polynomial f in F as input, where F=Fq is a finite field, we give a randomized algorithm that runs in time polynomial in s, n and log q that computes a factorization of the polynomial f as a product f=f1f2...fr, where each fi is an irreducible polynomial that is output as a noncommutative algebraic branching program. The algorithm works by first transforming the given algebraic branching program computing f into a linear matrix L using Higman's linearization of polynomials. We then factorize the linear matrix L and recover the factorization of the polynomial f. We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields.