In this paper we study the complexity of factorization of polynomials in the free noncommutative ring F x 1 x 2 x n of polynomials over the field F and noncommuting variables x 1 x 2 x n . Our main results are the following.

Although F x 1 x n is not a unique factorization ring, we note that variable-disjoint factorization in F x 1 x n 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 of Kaltofen and Trager [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 F x 1 x n which is a natural generalization of homogeneous polynomial factorization and prove some complexity bounds for it.