A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results

Vikraman Arvind; Pushkar S Joglekar

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arXiv · 2404.16382

A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results

Vikraman Arvind·Pushkar S Joglekar

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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 \cite{Co90,Cohnfir} 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.

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Vikraman Arvind, Pushkar S Joglekar. 2024-04-25. A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results. https://arxiv.org/abs/2404.16382

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