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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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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.

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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

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.

Subject Classification

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

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