A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results

TL;DR

This paper explores the relationship between noncommutative rank and rational identity testing, establishing reductions that suggest solving bivariate cases could lead to deterministic parallel algorithms for the general problems.

Contribution

It introduces deterministic NC reductions from multivariate to bivariate noncommutative rank and rational identity testing, linking their computational complexities.

Findings

Deterministic NC reductions from multivariate to bivariate ncRANK and RIT.

Bivariate RIT reduces to bivariate ncRANK in NC.

Implication that a deterministic NC algorithm for bivariate ncRANK would solve multivariate cases.

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.