Density of Complex Zeros of a System of Real Random Polynomials

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 $m\ge 2$ case than in the $m=1$ case: the density goes to infinity instead of tending linearly to zero.