On the Complexity of Noncommutative Polynomial Factorization

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.