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On Efficient Noncommutative Polynomial Factorization via Higman Linearization

Authors Vikraman Arvind, Pushkar S. Joglekar

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Author Details

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

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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)


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.

Subject Classification

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


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  1. S.A Amitsur. Rational identities and applications to algebra and geometry. Journal of Algebra, 3(3):304-359, 1966. URL:
  2. S.A. Amitsur and J. Levitzki. Minimal identities for algebras. Proceedings of the American Mathematical Society, 4(2):449-463, 1950. Google Scholar
  3. V. Arvind, Pushkar S. Joglekar, and Gaurav Rattan. On the complexity of noncommutative polynomial factorization. Inf. Comput., 262:22-39, 2018. URL:
  4. Vikraman Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. A special case of rational identity testing and the brešar-klep theorem. In Javier Esparza and Daniel Král', editors, 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic, volume 170 of LIPIcs, pages 10:1-10:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  5. Andrej Bogdanov and Hoeteck Wee. More on noncommutative polynomial identity testing. In 20th Annual IEEE Conference on Computational Complexity (CCC 2005), 11-15 June 2005, San Jose, CA, USA, pages 92-99, 2005. URL:
  6. W. Burnside. On the condition of reducibility of any group of linear substitutions. Proceedings of London Mathematical Society, 3:430-434, 1905. Google Scholar
  7. P. M. Cohn. Free Ideal Rings and Localization in General Rings. New Mathematical Monographs. Cambridge University Press, 2006. URL:
  8. P. M. Cohn. Introduction To Ring Theory. Springer, 2011. Google Scholar
  9. Richard A. DeMillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978. URL:
  10. Harm Derksen and Visu Makam. Polynomial degree bounds for matrix semi-invariants. Advances in Mathematics, 310:44-63, 2017. URL:
  11. Michael A. Forbes. Polynomial identity testing of read-once oblivious algebraic branching programs. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2014. URL:
  12. Katalin Friedl and Lajos Rónyai. Polynomial time solutions of some problems in computational algebra. In Robert Sedgewick, editor, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, May 6-8, 1985, Providence, Rhode Island, USA, pages 153-162. ACM, 1985. URL:
  13. Ankit Garg, Leonid Gurvits, Rafael Mendes de Oliveira, and Avi Wigderson. Operator scaling: Theory and applications. Found. Comput. Math., 20(2):223-290, 2020. URL:
  14. J Helton, Igor Klep, and Jurij Volčič. Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra. International Mathematics Research Notices, 2022(1):343-372, June 2020. URL:
  15. Graham Higman. The units of group-rings. Proceedings of the London Mathematical Society, s2-46(1):231-248, 1940. URL:
  16. Pavel Hrubes and Avi Wigderson. Non-commutative arithmetic circuits with division. Theory Comput., 11:357-393, 2015. URL:
  17. Gábor Ivanyos, Youming Qiao, and K. V. Subrahmanyam. Non-commutative edmonds' problem and matrix semi-invariants. Comput. Complex., 26(3):717-763, 2017. URL:
  18. Gábor Ivanyos, Youming Qiao, and K. V. Subrahmanyam. Constructive non-commutative rank computation is in deterministic polynomial time. Comput. Complex., 27(4):561-593, 2018. URL:
  19. Erich Kaltofen. Factorization of polynomials given by straight-line programs. Adv. Comput. Res., 5:375-412, 1989. Google Scholar
  20. Erich Kaltofen and Barry M. Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symb. Comput., 9(3):301-320, 1990. URL:
  21. Swastik Kopparty, Shubhangi Saraf, and Amir Shpilka. Equivalence of polynomial identity testing and polynomial factorization. Comput. Complex., 24(2):295-331, 2015. URL:
  22. Noam Nisan. Lower bounds for non-commutative computation (extended abstract). In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 410-418, 1991. URL:
  23. Ran Raz and Amir Shpilka. Deterministic polynomial identity testing in non-commutative models. Computational Complexity, 14(1):1-19, 2005. URL:
  24. Lajos Rónyai. Computing the structure of finite algebras. J. Symb. Comput., 9(3):355-373, 1990. URL:
  25. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. URL:
  26. Joachim von zur Gathen and Jürgen Gerhard. Modern Computer Algebra (3. ed.). Cambridge University Press, 2013. Google Scholar
  27. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Edward W. Ng, editor, Symbolic and Algebraic Computation, EUROSAM '79, An International Symposiumon Symbolic and Algebraic Computation, Marseille, France, June 1979, Proceedings, volume 72 of Lecture Notes in Computer Science, pages 216-226. Springer, 1979. URL:
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