Central limit theorem for the number of real roots of Kostlan Shub Smale random polynomial systems.

The roots of random polynomials and of random polynomial systems have been extensively studied in the past. While the picture is quite complete in the case of polynomials, the case of systems is much harder. Kostlan, Shub, and Smale random polynomial systems were introduced in the nineties and the expectation of the number of their real roots is known since then. Only recently the asymptotic variance, as the degree goes to infinity, was obtained. In the present paper we obtain a Central Limit Theorem for the number of real roots of a square Kostlan-Shub-Smale system of any size as the degree goes to infinity.