Central Limit Theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems

TL;DR

This paper proves a Central Limit Theorem for the normalized volume of the zero set of high-degree Kostlan-Shub-Smale random polynomial systems, extending previous results on the number of real roots.

Contribution

It extends the CLT to the volume of zero sets for rectangular systems using Kac-Rice and Wiener Chaos techniques, generalizing prior square case results.

Findings

CLT established for the volume of zero sets as degree increases

Uses Kac-Rice formula for second moments

Employs Itô-Wiener Chaos expansion

Abstract

We state 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 {\it Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems} by the same authors in which the square case was considered. Our main tools are the Kac-Rice formula for the second moment of the volume of the zero set and an expansion of this random variable into the It\^o-Wiener Chaos.