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A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results

Authors Vikraman Arvind , Pushkar S. Joglekar

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • noncommutative rank
  • rational formulas
  • identity testing
  • parallel complexity

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Abstract

We study the noncommutative rank problem, ncRANK, of computing the rank of matrices with linear entries in n noncommuting variables and the problem of noncommutative Rational Identity Testing, RIT, which is to decide if a given rational formula in n noncommuting variables is zero on its domain of definition.
Motivated by the question whether these problems have deterministic NC algorithms, we revisit their interrelationship from a parallel complexity point of view. We show the following results:  
1) Based on Cohn’s embedding theorem [Cohn, 1990; Cohn, 2006] we show deterministic NC reductions from multivariate ncRANK to bivariate ncRANK and from multivariate RIT to bivariate RIT. 
2) We obtain a deterministic NC-Turing reduction from bivariate RIT to bivariate ncRANK, thereby proving that a deterministic NC algorithm for bivariate ncRANK would imply that both multivariate RIT and multivariate ncRANK are in deterministic NC.

Cite As Get BibTex

Vikraman Arvind and Pushkar S. Joglekar. A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.14

Author Details

Vikraman Arvind
  • The Institute of Mathematical Sciences (HBNI), Chennai, India
  • Chennai Mathematical Institute, Siruseri, Kelambakkam, India
Pushkar S. Joglekar
  • Vishwakarma Institute of Technology, Pune, India

Acknowledgements

We are grateful to the anonymous referees for their valuable comments and suggestions.

References

  1. S. A. Amitsur. Rational identities and applications to algebra and geometry. Journal of Algebra, 3:304-359, 1966. Google Scholar
  2. V. Arvind, Abhranil Chatterjee, and Partha Mukhopadhyay. Black-box identity testing of noncommutative rational formulas in deterministic quasipolynomial time. CoRR, abs/2309.1564, 2023. In STOC 2024, to appear. URL: https://arxiv.org/abs/2309.15647.
  3. V. Arvind, Abhranil Chatterjee, and Partha Mukhopadhyay. Trading determinism for noncommutativity in Edmonds' problem. CoRR, abs/2404.07986, 2024. URL: https://arxiv.org/abs/2404.07986.
  4. V. Arvind and Pushkar Joglekar. A multivariate to bivariate reduction for noncommutative rank and related results. CoRR, abs/2404.16382, 2024. URL: https://arxiv.org/abs/2404.16382.
  5. Vikraman Arvind, Pushkar S. Joglekar, Partha Mukhopadhyay, and S. Raja. Randomized polynomial-time identity testing for noncommutative circuits. Theory Comput., 15:1-36, 2019. URL: https://doi.org/10.4086/TOC.2019.V015A007.
  6. G. M. Bergman. Skew fields of noncommutative rational functions, after Amitsur. Sé Schü–Lentin–Nivat, Paris, 1970. Google Scholar
  7. Markus Bläser, Gorav Jindal, and Anurag Pandey. A deterministic PTAS for the commutative rank of matrix spaces. Theory Comput., 14(1):1-21, 2018. URL: https://doi.org/10.4086/TOC.2018.V014A003.
  8. 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: https://doi.org/10.1109/CCC.2005.13.
  9. Richard P. Brent. The parallel evaluation of general arithmetic expressions. J. ACM, 21(2):201-206, 1974. URL: https://doi.org/10.1145/321812.321815.
  10. Abhranil Chatterjee and Partha Mukhopadhyay. The noncommutative edmonds' problem re-visited. CoRR, abs/2305.09984, 2023. URL: https://doi.org/10.48550/arXiv.2305.09984.
  11. P. M. Cohn. The embedding of fir in skew fields. Proceedings of the London Mathematical Society, 23:193-213, 1971. Google Scholar
  12. P. M. Cohn. Universal skew fields of fractions. Sympos. Math., 8:135-148, 1972. Google Scholar
  13. P. M. Cohn. Free Rings and their Relations. London Mathematical Society Monographs. Academic Press, 1985. Google Scholar
  14. P. M. Cohn. An embedding theorem for free associative algebras. Mathematica Pannonica, 1(1):49-56, 1990. Google Scholar
  15. P. M. Cohn. Free Ideal Rings and Localization in General Rings. New Mathematical Monographs. Cambridge University Press, 2006. URL: https://doi.org/10.1017/CBO9780511542794.
  16. Richard A. DeMillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978. URL: https://doi.org/10.1016/0020-0190(78)90067-4.
  17. Harm Derksen and Visu Makam. Polynomial degree bounds for matrix semi-invariants. CoRR, abs/1512.03393, 2015. URL: https://arxiv.org/abs/1512.03393.
  18. Ankit Garg, Leonid Gurvits, Rafael Mendes de Oliveira, and Avi Wigderson. A deterministic polynomial time algorithm for non-commutative rational identity testing. CoRR, abs/1511.03730, 2015. URL: https://arxiv.org/abs/1511.03730.
  19. 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: https://doi.org/10.1007/s10208-019-09417-z.
  20. Graham Higman. The units of group-rings. Proceedings of the London Mathematical Society, s2-46(1):231-248, 1940. URL: https://doi.org/10.1112/plms/s2-46.1.231.
  21. Pavel Hrubes and Avi Wigderson. Non-commutative arithmetic circuits with division. Theory Comput., 11:357-393, 2015. URL: https://doi.org/10.4086/TOC.2015.V011A014.
  22. Gábor Ivanyos, Youming Qiao, and K. V. Subrahmanyam. Non-commutative edmonds' problem and matrix semi-invariants. Comput. Complex., 26(3):717-763, 2017. URL: https://doi.org/10.1007/s00037-016-0143-x.
  23. 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: https://doi.org/10.1007/s00037-018-0165-7.
  24. 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. ACM, 1991. URL: https://doi.org/10.1145/103418.103462.
  25. Sanguthevar Rajasekaran and John H. Reif, editors. Handbook of Parallel Computing - Models, Algorithms and Applications. Chapman and Hall/CRC, 2007. URL: https://doi.org/10.1201/9781420011296.
  26. Ran Raz and Amir Shpilka. Deterministic polynomial identity testing in non-commutative models. Comput. Complex., 14(1):1-19, 2005. URL: https://doi.org/10.1007/S00037-005-0188-8.
  27. L. H. Rowen. Polynomial identities in ring theory, Pure and Applied Mathematics. Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York, 1980. Google Scholar
  28. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. URL: https://doi.org/10.1145/322217.322225.
  29. James C. Wyllie. Complexity of Parallel Computation, PhD Thesis. Cornell University, 1979. Google Scholar
  30. 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: https://doi.org/10.1007/3-540-09519-5_73.
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