Concentration for the zero set of large random polynomial systems

For random systems of $K$ polynomials in $N+1$ real variables which include the models of Kostlan (1987) and Shub and Smale (1993), we prove that the number of zeros on the unit sphere for $K=N$ or the Hausdorff measure of the zero set for $K<N$ concentrates around its mean as $N\to \mathrm{\infty }$. 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 set of parameters which determine the variance of their Gaussian coefficients. We prove that the convergence is uniform in those parameters and $K$.