  For random systems of $K$ polynomials in $N + 1$ real variables which include
the models of Kostlan (1987) and Shub and Smale (1992), we prove that the
number of zeros for $K = N$ or the volume of the zero set for $K < N$ on the
unit sphere concentrates around its mean as $N\to\infty$. To prove
concentration we show that the variance of the latter random variable
normalized by its mean goes to zero. The polynomial systems we consider depend
on a choice of a set of parameters which determine the variance of their
Gaussian coefficients. We prove that the convergence is uniform in those
parameters and K.

Probability