On distribution of zeros of random polynomials in complex plane

TL;DR

This paper studies the distribution of roots of random polynomials with i.i.d. coefficients, showing uniform angular distribution and conditions for roots to concentrate near the unit circle as degree increases.

Contribution

It establishes the asymptotic uniform distribution of root arguments and identifies the necessary and sufficient condition for roots to cluster near the unit circle.

Findings

Arguments of roots are uniformly distributed asymptotically.

Finite expected logarithm of coefficient magnitude is necessary and sufficient for roots to concentrate near the unit circle.

Provides conditions for the asymptotic behavior of roots in random polynomials.

Abstract

Let $G_n(z)=\xi_0+\xi_1z+...+\xi_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2\pi]$ asymptotically as $n\to\infty$. We also prove that the condition $\E\ln(1+|\xi_0|)<\infty$ is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.