Expected discrepancy for zeros of random algebraic polynomials

TL;DR

This paper investigates how the zeros of random algebraic polynomials tend to cluster and how their expected discrepancy decreases as the degree increases, providing new insights into their distribution patterns.

Contribution

It introduces a novel analysis of the expected discrepancy decay rate for polynomial roots with possibly dependent coefficients, extending previous results to broader settings.

Findings

Expected discrepancy decays like √(log n / n)

Zeros tend to cluster near the unit circle

Number of zeros in specific regions analyzed

Abstract

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree $n$, with not necessarily independent coefficients, decays like $\sqrt{\log n/n}$. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of the plane, such as circles centered on the unit circumference, and polygons inscribed in the unit circumference.