CLT for the zeros of Kostlan Shub Smale random polynomials

TL;DR

This paper establishes the asymptotic behavior of the variance and proves a central limit theorem for the number of roots of Kostlan-Shub-Smale random polynomials as their degree increases.

Contribution

It provides the first detailed asymptotic analysis and CLT for the zeros of Kostlan-Shub-Smale polynomials.

Findings

Asymptotic main term of variance derived

Central limit theorem proved for the number of roots

Results hold as polynomial degree tends to infinity

Abstract

In this paper we find the asymptotic main term of the variance of the number of roots of Kostlan-Shub-Smale random polynomials and prove a central limit theorem for the number of roots as the degree goes to infinity.