On Efficient Noncommutative Polynomial Factorization via Higman Linearization

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.