  The identity testing of rational formulas (RIT) in the free skew field
efficiently reduces to computing the rank of a matrix whose entries are linear
polynomials in noncommuting variables\cite{HW15}. This rank computation problem
has deterministic polynomial-time white-box algorithms \cite{GGOW16, IQS18} and
a randomized polynomial-time algorithm in the black-box setting \cite{DM17}. In
this paper, we propose a new approach for efficient derandomization of
\emph{black-box} RIT. Additionally, we obtain results for matrix rank
computation over the free skew field, and construct efficient linear pencil
representations for a new class of rational expressions. More precisely, we
show the following results:
  1. Under the hardness assumption that the ABP (algebraic branching program)
complexity of every polynomial identity for the $k\times k$ matrix algebra is
$2^{\Omega(k)}$ \cite{BW05}, we obtain a subexponential-time black-box
algorithm for RIT in almost general setting. This can be seen as the first
"hardness implies derandomization" type theorem for rational formulas.
  2. We show that the noncommutative rank of any matrix over the free skew
field whose entries have small linear pencil representations can be computed in
deterministic polynomial time. Prior to this, an efficient rank computation was
only known for matrices with noncommutative formulas as entries\cite{GGOW20}.
As special cases of our algorithm, we obtain the first deterministic
polynomial-time algorithms for rank computation of matrices whose entries are
noncommutative ABPs or rational formulas.
  3. Motivated by the definition given by Bergman\cite{Ber76}, we define a new
class that contains noncommutative ABPs and rational formulas. We obtain a
polynomial-size linear pencil representation for this class. As a by-product,
we obtain a white-box deterministic polynomial-time identity testing algorithm
for the class.
