Real roots of random polynomials: expectation and repulsion

TL;DR

This paper establishes optimal bounds on real roots repulsion and the probability of double roots for Kac random polynomials, and precisely determines the expected number of real roots for a broad class of coefficient distributions, extending previous Gaussian-only results.

Contribution

It provides the first optimal bounds on roots repulsion and double root probability for general iid coefficients, and extends the expected real roots formula beyond Gaussian cases.

Findings

Optimal bounds on real roots repulsion.

Bound on probability of double roots.

Expected number of real roots is (2/π) log n + C + o(1).

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.