Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems

TL;DR

This paper proves a Central Limit Theorem for the normalized count of real roots in large-degree Kostlan-Shub-Smale random polynomial systems, using Wiener chaos and the Fourth Moment Theorem.

Contribution

It introduces a novel approach to establish asymptotic normality for the number of roots via Wiener chaos representation and variance analysis.

Findings

Asymptotic normality of the number of real roots as degree increases

Explicit Wiener chaos representation of the root count

Analysis of the variance leading to CLT application

Abstract

We obtain a Central Limit Theorem for the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size as the degree goes to infinity. A study of the asymptotic variance of the number of roots is needed, this result was obtained in [2]. Afterwards we represent the number of roots as an explicit non linear functional belonging to the It{\^o}-Wiener chaos. This representation provides a tool for applying the Fourth Moment Theorem and henceforth the asymptotic normality. MSC 2010 subject classifications: Primary 60F05, 30C15, ; secondary 60G60, 65H10.