Zero Distribution of Random Polynomials

TL;DR

This paper investigates the global distribution of zeros for various ensembles of random polynomials, providing both almost sure convergence results and quantitative estimates on zero distributions.

Contribution

It extends known results on zero distribution to polynomials with dependent and non-identically distributed coefficients, and offers new quantitative discrepancy estimates.

Findings

Zeros of Kac polynomials are asymptotically uniformly distributed near the unit circle.

Expected discrepancy between zero measures and arclength measure is quantitatively estimated.

Almost sure convergence of zero measures to equilibrium measures is established and quantified.

Abstract

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that their zeros are asymptotically uniformly distributed near the unit circumference under mild assumptions on the coefficients. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane, and quantify this convergence. Random coefficients may be dependent and need not have identical distributions in our results.