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.

翻译：我们研究非交换秩问题（ncRANK），即计算具有$n$个非交换变量线性项矩阵的秩，以及非交换有理恒等式检验问题（RIT），即判定以$n$个非交换变量表达的有理公式在其定义域上是否为零。受这些问题是否存在确定性NC算法的启发，我们从并行复杂性的视角重新审视它们之间的相互关系。我们得到以下结果：1. 基于Cohn嵌入定理\cite{Co90,Cohnfir}，我们给出了从多变量ncRANK到双变量ncRANK以及从多变量RIT到双变量RIT的确定性NC约简；2. 我们获得了从双变量RIT到双变量ncRANK的确定性NC-图灵约简，从而证明若存在双变量ncRANK的确定性NC算法，则多变量RIT和多变量ncRANK均属于确定性NC。