Zeros of Complex Random Polynomials Spanned by Bergman Polynomials

TL;DR

This paper derives explicit formulas and asymptotic results for the expected number of zeros of complex Gaussian random polynomials spanned by orthogonal polynomials on the unit disk, revealing a universal zero distribution pattern.

Contribution

It provides explicit formulas for zero counts of Gaussian random polynomials with orthogonal bases and extends results to general orthogonal polynomial systems with regularity conditions.

Findings

Expected zeros in the unit disk are approximately 2n/3.

Explicit formulas for zeros in disks of radius r<1.

Asymptotic zero distribution is universal under regularity conditions.

Abstract

We study the expected number of zeros of $$P_n(z)=\sum_{k=0}^n\eta_kp_k(z),$$ where $\{\eta_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ are polynomials orthogonal on the unit disk. When $p_k(z)=\sqrt{(k+1)/\pi} z^k$, $k\in \{0,1,\dots, n\}$, we give an explicit formula for the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is $2n/3$. Generalizing our basis functions $\{p_k(z)\}$ to be regular in the sense of Ullman--Stahl--Totik, and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros of $P_n(z)$ in a disk of radius $r\in (0,1)$ centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is $2n/3$.