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.

49 pages, 2 figures, 1 table, final version

References in corpus (2)

• Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation
• Non universality for the variance of the number of real roots of random trigonometric polynomials

Cited by in corpus (1)

• Real roots of random polynomials: asymptotics of the variance