On the complexity of noncommutative polynomial factorization

Abstract

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 variable-disjoint 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 variable-disjoint factorization in the black-box setting can be efficiently computed (where the computed factors will be also given by black-boxes).•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.