Real roots of random polynomials: asymptotics of the variance

Abstract

We compute the precise leading asymptotics of the variance of the number of real roots for a large class of random polynomials, where the random coefficients have polynomial growth. Our results apply to many classical ensembles, including the Kac polynomials, hyperbolic polynomials, their derivatives, and any linear combinations of these polynomials. Prior to this paper, such asymptotics were established only for the Kac polynomials in the 1970s, with the seminal contribution of Maslova. The main ingredients of the proof are new asymptotic estimates for the two-point correlation function of the real roots, revealing geometric structures in the distribution of the real roots of these random polynomials. As a corollary, we obtain asymptotic normality for the real roots of these random polynomials, extending and strengthening a related result of O. Nguyen and V. Vu.

37 pages, 6 figures, final version

References in corpus (4)

• Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation
• Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation
• Non universality for the variance of the number of real roots of random trigonometric polynomials
• The number of real zeros of elliptic polynomials