Central limit theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems.

We establish the Central Limit Theorem, as the degree goes to infinity, for the normalized volume of the zero set of a rectangular Kostlan–Shub–Smale random polynomial system. This paper is a continuation of Central Limit Theorem for the number of real roots of Kostlan–Shub–Smale random polynomial systems by the same authors in which the case of square systems was considered. Our main tools are Kac-Rice formula and an expansion of the volume of the level set into the Itˆo-Wiener Chaos. Keywords: Kostlan–Shub–Smale random polynomial systems, co-area formula, Kac-Rice formula, central limit theorem. AMS sujet classification: Primary: 60F05, 30C15. Secondary: 60G60, 65H10