# Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications

## TL;DR

This paper introduces a comparison principle to analyze the distribution of real roots in random polynomials with coefficients of polynomial growth, extending results beyond zero-mean cases and applying to various polynomial classes.

## Contribution

It develops a novel comparison principle that reduces the analysis of non-centered coefficients to the mean-zero case, enabling new results for diverse random polynomial models.

## Key findings

- New estimates for the number of real roots in various polynomial classes
- Logarithmic integrability estimates for random polynomials
- Sharp local estimates for real zeros

## Abstract

This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number of real zeros. Almost all previous results require coefficients with zero mean, and it is non-trivial to extend these results to the general case. Our approach is based on a novel comparison principle that reduces the general situation to the mean-zero setting. As applications, we obtain new results for the Kac polynomials, hyperbolic random polynomials, their derivatives, and generalizations of these polynomials. The proof features new logarithmic integrability estimates for random polynomials (both local and global) and fairly sharp estimates for the local number of real zeros.