On the Complexity of Noncommutative Polynomial Factorization

TL;DR

This paper investigates the complexity of factoring noncommutative polynomials, establishing polynomial-time equivalences with Polynomial Identity Testing and demonstrating efficient algorithms for certain classes of polynomials.

Contribution

It introduces the concept of variable-disjoint factorization in noncommutative polynomials, proves its polynomial-time equivalence to PIT, and provides efficient algorithms for specific polynomial classes.

Findings

Variable-disjoint factorization is unique in noncommutative rings.

Factorization in this setting is polynomial-time equivalent to Polynomial Identity Testing.

Homogeneous and multilinear noncommutative polynomials have efficiently computable unique factorizations.

Abstract

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring $\mathbb{F}\langle x_1,x_2,\dots,x_n\rangle$ of polynomials over the field $\mathbb{F}$ and noncommuting variables $x_1,x_2,\ldots,x_n$. Our main results are the following. Although $\mathbb{F}\langle x_1,x_2,\dots,x_n \rangle$ is not a unique factorization ring, we note that variable-disjoint factorization in $\mathbb{F}\langle x_1,x_2,\dots,x_n \rangle$ has the uniqueness property. Furthermore, we prove that computing the variable-disjoint factorization is polynomial-time equivalent to Polynomial Identity Testing (both when the input polynomial is given by an arithmetic circuit or an algebraic branching program). We also show that variable-disjoint factorization in the black-box setting can be efficiently computed (where the factors computed will be also given by black-boxes, analogous to the work [KT91] in the commutative setting). As a consequence of the previous result we show that homogeneous noncommutative polynomials and multilinear noncommutative polynomials have unique factorizations in the usual sense, which can be efficiently computed. Finally, we discuss a polynomial decomposition problem in $\mathbb{F}\langle x_1,x_2,\dots,x_n\rangle$ which is a natural generalization of homogeneous polynomial factorization and prove some complexity bounds for it.