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DOI: 10.4230/LIPIcs.CCC.2022.12
URN: urn:nbn:de:0030-drops-165747
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16574/
Arvind, Vikraman ; Joglekar, Pushkar S.
On Efficient Noncommutative Polynomial Factorization via Higman Linearization
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Abstract
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.
BibTeX - Entry
@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/opus/volltexte/2022/16574},
URN = {urn:nbn:de:0030-drops-165747},
doi = {10.4230/LIPIcs.CCC.2022.12},
annote = {Keywords: Noncommutative Polynomials, Arithmetic Circuits, Factorization, Identity testing}
}
| Keywords: | Noncommutative Polynomials, Arithmetic Circuits, Factorization, Identity testing | |
| Seminar: | 37th Computational Complexity Conference (CCC 2022) | |
| Issue date: | 2022 | |
| Date of publication: | 11.07.2022 |
