  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.
