Motivated by the Hadamard product of matrices we define the Hadamard

product of multivariate polynomials and study its arithmetic circuit

and branching program complexity. We also give applications and

connections to polynomial identity testing. Our main results are

the following.

\begin{itemize}

\item[$\bullet$] We show that noncommutative polynomial identity testing for

algebraic branching programs over rationals is complete for

the logspace counting class $\ceql$, and over fields of characteristic

$p$ the problem is in $\ModpL/\Poly$.

\item[$\bullet$] We show an exponential lower bound for expressing the

Raz-Yehudayoff polynomial as the Hadamard product of two monotone

multilinear polynomials. In contrast the Permanent can be expressed

as the Hadamard product of two monotone multilinear formulas of

quadratic size.

\end{itemize}