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On Efficient Noncommutative Polynomial Factorization via Higman Linearization

Authors Vikraman Arvind, Pushkar S. Joglekar

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Noncommutative Polynomials
  • Arithmetic Circuits
  • Factorization
  • Identity testing

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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.

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Vikraman Arvind and Pushkar S. Joglekar. On Efficient Noncommutative Polynomial Factorization via Higman Linearization. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.12

Author Details

Vikraman Arvind
  • Institute of Mathematical Sciences, Chennai, India
Pushkar S. Joglekar
  • Vishwakarma Institute of Technology, Pune, India

Funding

  • Joglekar, Pushkar S.: Author would like to thank Science and Engineering Research Board for the funding through the MATRICS project, File no. MTR/2018/001214.

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