Real roots of random polynomials: expectation and repulsion

Abstract

Let $P_{n}(x)= \sum_{i=0}^n ξ_i x^i$ be a Kac random polynomial where the coefficients $ξ_i$ are iid 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 double root. As an application, we consider the problem of estimating the number of real roots of $P_n$, 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 $P_n(x)$ is exactly $\frac{2}π \log n +C +o(1)$, 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.