Real Roots of Random Polynomials: Expectation and Repulsion

Let Pn(x) = ∑n i=0 ξix i be a Kac random polynomial where the coefficients ξi 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 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 logn + 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.