The number of real zeros of elliptic polynomials

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.