Real roots of random polynomials: expectation and repulsion

Abstract

Let $P_n(x)={\sum }_{i=0}^n{\xi }_i{x}^i$ be a Kac random polynomial where the coefficients ${\xi }_i$ are iid copies of a given random variable $\xi$. 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 $\xi$, that the expected number of real roots of $P_n(x)$ is exactly $\frac{2}{\pi }\log n+C+o(1)$, where $C$ is an absolute constant depending on the atom variable $\xi$. Prior to this paper, such a result was known only for the case when $\xi$ is Gaussian.

31 pages, 2 figures

References in corpus (1)

• A Numerical Algorithm for Zero Counting. III: Randomization and Condition

Cited by in corpus (3)

• Double roots of random polynomials with integer coefficients
• The number of real zeros of elliptic polynomials
• Real roots of random polynomials: asymptotics of the variance