  We study the density of complex zeros of a system of real random SO($m+1$)
polynomials in several variables. We show that the density of complex zeros of
this random polynomial system with real coefficients rapidly approaches the
density of complex zeros in the complex coefficients case. We also show that
the behavior the scaled density of complex zeros near $\mathbb{R}^m$ of the
system of real random polynomials is different in the $m\geq 2$ case than in
the $m=1$ case: the density goes to infinity instead of tending linearly to
zero.
