# Random polynomials: central limit theorems for the real roots

## TL;DR

This paper establishes a new, more general central limit theorem for the number of real roots in a broad class of random polynomials, extending Maslova's classical result from 1974.

## Contribution

It introduces a novel approach that generalizes and strengthens the existing CLT for real roots of random polynomials with polynomially growing coefficients.

## Key findings

- Derives a general CLT for real roots of random polynomials
- Extends Maslova's theorem to broader classes of polynomials
- Provides a new methodology for analyzing real roots

## Abstract

The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood-Offord and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova's theorem.