License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2022.12
URN: urn:nbn:de:0030-drops-165747
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Arvind, Vikraman ; Joglekar, Pushkar S.

On Efficient Noncommutative Polynomial Factorization via Higman Linearization

LIPIcs-CCC-2022-12.pdf (0.8 MB)


In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring ?∠{x_1,x_2,…,x_n} of polynomials in noncommuting variables x_1,x_2,…,x_n over the field ?. 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 ∈ ?∠{x_1,x_2,…,x_n}, where ? = ?_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

  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 =		{},
  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
Collection: 37th Computational Complexity Conference (CCC 2022)
Issue Date: 2022
Date of publication: 11.07.2022

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