Roots of random polynomials with coefficients of polynomial growth

Abstract

In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.

Expected number of real zeroes of random Taylor series

“…It seems that only very little is known about real zeroes of random Taylor series. One exception is the paper of Do et al [10,Section 2.5] who proved a local universality result for real and complex zeroes. Our approach is a development of the method of Ibragimov and Maslova [15] and is independent of the method of [10].…”

Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature

“…Further results on the number of real zeroes, including an asymptotic formula for the variance and a central limit theorem, were obtained in the subsequent works by Ibragimov and Maslova [23,35,34]. For more recent results on the number of real roots, see [10,36,11].…”

“…Proof. By (11), N n [t, t + δ/ √ n] is the number of real zeroes of Q n,t (·) in the interval [0, δ]. By Theorem 2.2, the latter process converges weakly to Z γ(t) (·) on the space A real (D R ), as n → ∞.…”

Real zeros of random sums with i.i.d. coefficients

“…random variables, and {c j } are deterministic weights. For results concerning the weighted random polynomial G n we direct the reader to the works [10], [35], and [9]. Appealing to Fourier transforms of distribution functions, Bleher and Di [3] gave a universality result for the expected number of real zeros of G n defined in (1.3) when coefficients {c i } are elliptical weights, that is weights of the form n i , and {η i } are i.i.d.…”