On Efficient Noncommutative Polynomial Factorization via Higman Linearization
V. Arvind, Pushkar S. Joglekar
TL;DR
This work addresses the problem of factorizing noncommutative polynomials over finite fields when the input is given by an algebraic branching program (ABP). It introduces a three-phase pipeline: first, Higman linearization converts a noncommutative polynomial $f$ into a linear matrix $L$; second, Ronyai’s common invariant-subspace algorithm (together with Cohn’s free ideal ring theory) factors $L$ into atomic linear factors; third, a linear-algebraic extraction recovers the irreducible factors $f_i$ of $f$ from the factors of $L$. The main result is a randomized polynomial-time algorithm over finite fields that outputs each irreducible factor as an ABP, with deterministic polynomial-time factorization possible for small fields; the approach also handles commutatively zero polynomials via dilations and specialized lemmas. The work connects noncommutative factorization with invariant-subspace computation, Higman linearization, and noncommutative algebraic structures, and clarifies the limitations for rational (over $\mathbb{Q}$) factorization. Overall, it substantially advances efficient noncommutative factorization in the ABP/finite-field setting and maps clear avenues for future work on rational numbers and deterministic regimes.
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.