On the Complexity of Noncommutative Polynomial Factorization

Abstract

AbstractIn this paper we study the complexity of factorization of polynomials in the free noncommutative ring \(\mathbb {F}\langle x_1,x_2,\ldots ,x_n \rangle \) of polynomials over the field \(\mathbb {F}\) and noncommuting variables \(x_1,x_2,\ldots ,x_n\). Our main results are the following: Although \(\mathbb {F}\langle x_1,\ldots ,x_n \rangle \) is not a unique factorization ring, we note that variable-disjoint factorization in \(\mathbb {F}\langle x_1,\ldots ,x_n \rangle \) 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 [12] 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 \(\mathbb {F}\langle x_1,\ldots ,x_n \rangle \) which is a natural generalization of homogeneous polynomial factorization and prove some complexity bounds for it. KeywordsHomogeneous PolynomialUnique FactorizationArithmetic CircuitPolynomial FactorizationMultilinear PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.