On the Complexity of Noncommutative Polynomial Factorization

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring F〈x1, x2, . . . , xn〉 of polynomials over the field F and noncommuting variables x1, x2, . . . , xn. Our main results are the following: • Although F〈x1, . . . , xn〉 is not a unique factorization ring, we note that variabledisjoint factorization in F〈x1, . . . , xn〉 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 variabledisjoint 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 [KT90] 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 F〈x1, . . . , xn〉 which is a natural generalization of homogeneous polynomial factorization and prove some complexity bounds for it.