In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring 𝔽∠{x_1,x_2,…,x_n} of polynomials in noncommuting variables x_1,x_2,…,x_n over the field 𝔽. We obtain the following result: - We give a randomized algorithm that takes as input a noncommutative arithmetic formula of size s computing a noncommutative polynomial f ∈ 𝔽∠{x_1,x_2,…,x_n}, where 𝔽 = 𝔽_q is a finite field, and in time polynomial in s, n and log₂q computes a factorization of f as a product f = f_1f_2 ⋯ f_r, where each f_i is an irreducible polynomial that is output as a noncommutative algebraic branching program. - The algorithm works by first transforming f into a linear matrix L using Higman’s linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f. We use basic elements from Cohn’s theory of free ideals rings combined with Ronyai’s randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields.
@InProceedings{arvind_et_al:LIPIcs.CCC.2022.12, author = {Arvind, Vikraman and Joglekar, Pushkar S.}, title = {{On Efficient Noncommutative Polynomial Factorization via Higman Linearization}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {12:1--12:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.12}, URN = {urn:nbn:de:0030-drops-165747}, doi = {10.4230/LIPIcs.CCC.2022.12}, annote = {Keywords: Noncommutative Polynomials, Arithmetic Circuits, Factorization, Identity testing} }
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