The number of real zeros of elliptic polynomials

TL;DR

This paper derives precise statistical properties of the number of real zeros of Gaussian elliptic polynomials, including variance, asymptotics, and limit theorems, and extends the analysis to Gaussian analytic functions.

Contribution

It provides the first detailed variance formula, asymptotic expansions, and limit theorems for zeros of elliptic polynomials, extending to Gaussian analytic functions.

Findings

Exact variance formula for zeros of elliptic polynomials

Asymptotic expansion for large degree n

Conditions for CLT and law of large numbers

Abstract

Let $N_n(a, b)$ denote the number of real zeros of Gaussian elliptic polynomials of degree $n$ on the interval $(a, b)$, where $a$ and $b$ may vary with $n$. We obtain a precise formula for the variance of $N_n(a, b)$ and utilize this expression to derive an asymptotic expansion for large values of $n$. Furthermore, we provide sharp estimates for the cumulants and central moments of $N_n(a, b)$. These estimates are instrumental in establishing sufficient conditions on the interval $(a, b)$ for $N_n(a, b)$ to satisfy both a central limit theorem and a strong law of large numbers. In the second part of the paper, we extend our analysis to nondegenerate Gaussian analytic functions, including well-known examples such as the Gaussian Weyl series and Weyl polynomials.