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Erschienen in:
2015 | OriginalPaper | Buchkapitel
On the Complexity of Noncommutative Polynomial Factorization
verfasst von : V. Arvind, Gaurav Rattan, Pushkar Joglekar
Erschienen in: Mathematical Foundations of Computer Science 2015
Verlag: Springer Berlin Heidelberg
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Abstract
In 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:
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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).
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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.
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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.