The number of real zeros of elliptic polynomials

Nhan D. V. Nguyen

Published 2021-11-21, updated 2023-02-19Version 2

The aim of this paper is to further explore the number $N_n(a,b)$ of real zeros of elliptic polynomials of degree $n$ on an arbitrary interval $(a,b)$, where $a$ and $b$ may depend on $n$. We first develop an exact and accessible formula for the variance of $N_n(a,b)$ and exploit it to derive all terms in the large $n$ asymptotic expansion. We then provide sharp estimates for the cumulants and central moments of this quantity. These estimates play an important role in determining sufficient conditions on the interval $(a,b)$ under which $N_n(a,b)$ satisfies a central limit theorem and a strong law of large numbers.