  We study the density of complex critical points of a real random SO(m+1)
polynomial in m variables. In a previous paper [Mac09], the author used the
Poincare- Lelong formula to show that the density of complex zeros of a system
of these real random polynomials rapidly approaches the density of complex
zeros of a system of the corresponding complex random polynomials, the SU(m+1)
polynomials. In this paper, we use the Kac- Rice formula to prove an analogous
result: the density of complex critical points of one of these real random
polynomials rapidly approaches the density of complex critical points of the
corresponding complex random polynomial. In one variable, we give an exact
formula and a scaling limit formula for the density of critical points of the
real random SO(2) polynomial as well as for the density of critical points of
the corresponding complex random SU(2) polynomial.
