On efficient noncommutative polynomial factorization via higman linearization

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F〈 x 1 , x 2 ,..., x n 〉 of polynomials in noncommuting variables x 1 , x 2 ,..., x n over the field F. We obtain the following result: • We give a randomized algorithm that takes as input a noncommutative arithmetic formula of size s computing a noncommutative polynomial f ∈ F〈 x 1 , x 2 ,..., x n 〉, where F = F q is a finite field, and in time polynomial in s , n and log 2 q computes a factorization of f as a product f = f 1 f 2 ··· f r , where each f i is an irreducible polynomial that is output as a noncommutative algebraic branching program. • The algorithm works by first transforming f into a linear matrix L using Higman's linearization of polynomials. We then factorize the linear matrix L and recover the factorization of 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.