Real roots of random polynomials: expectation and repulsion

Abstract

Let Pn(x) = n i=0 ξix i be a Kac random polynomial where the coefficients ξi are i.i.d. copies of a given random variable ξ. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a real double root.As an application, we consider the problem of estimating the number of real roots of Pn, which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables ξ, that the expected number of real roots of Pn(x) is exactly (2/π) log n + C + o(1), where C is an absolute constant depending on the atom variable ξ. Prior to this paper, such a result was known only for the case when ξ is Gaussian.

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

“…This proves (15) and (16) in view of the formula for the covariance function of Z γ given in (10). It remains to verify the Lindeberg condition, namely…”

“…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].…”

Expected number of real zeroes of random Taylor series

“…Proof. Observe that ϕ → ϕ(1) defines a continuous mapping from H R (Q R ) to R. The continuous mapping theorem applied to (9), together with the asymptotics v(1−a) ∼ (2a) −γ L(1/a), a ↓ 0, that follows from Theorem 3.1, yields (10).…”

Double roots of random polynomials with integer coefficients

“…Then we have P P has a double root = P P has a double root at either 0, −1 or 1 + o(n −2 ) as n → ∞. (4) Moreover, if P(ξ 0 = 0) = 0, then P P has a double root = O(n −2 ).…”

“…The above bound follows from an application of optimal inverse Littlewood-Offord theorem [18,Theorem 2.5]. For details, see Lemma A.5 in [4] where the same has been proved under the assumption that ξ 0 has bounded (2 + ε) moment. This completes the proof of Theorem 1.1.…”