# On the asymptotic variance of the number of real roots of random polynomial systems

## TL;DR

This paper derives the asymptotic variance of the normalized count of real roots in high-degree Kostlan-Shub-Smale random polynomial systems, using Kac-Rice formula and Hermite expansions.

## Contribution

It provides the first explicit asymptotic variance formula for the number of real roots in these polynomial systems, extending understanding of their probabilistic behavior.

## Key findings

- Asymptotic variance formula derived for large degree
- Utilizes Kac-Rice formula for second factorial moment
- Employs Hermite expansion to analyze the variance

## Abstract

We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial moment of the number of roots and a Hermite expansion of this random variable.