Real roots of random polynomials: expectation and repulsion

Abstract

Let P n ( x ) = ∑ i = 0 n ξ i x i be a Kac random polynomial where the coefficients ξ i 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 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 ( 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.

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• Proceedings of the London Mathematical Society - View
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