Zeros of Polynomials with Random Coefficients

TL;DR

This paper investigates the distribution of zeros of random polynomials, providing quantitative estimates for their equidistribution near the unit circle, applicable to dependent coefficients and various bases.

Contribution

It offers new quantitative bounds on zero distribution for a broad class of random polynomials, including dependent coefficients and different bases.

Findings

Zeros are asymptotically equidistributed near the unit circle.

Provides bounds on expected discrepancy of zeros.

Analyzes polynomials with dependent and non-identically distributed coefficients.

Abstract

Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected number of roots in various sets. This is done for polynomials with coefficients that may be dependent, and need not have identical distributions. We also study random polynomials spanned by various deterministic bases.