Limit law for root separation in random polynomials

TL;DR

This paper investigates the asymptotic distribution of root separation distances in random polynomials with i.i.d. coefficients, revealing a convergence to a Poisson process and establishing a limit law for minimal separation.

Contribution

It proves the convergence of normalized root separation distances to a Poisson process and introduces a new result on the absence of double zeros for random Taylor series.

Findings

Normalized root separation distances converge to a Poisson process

Minimal root separation normalized by n^{-5/4} has a non-trivial limit law

Random Taylor series almost surely do not have double zeros except at the origin

Abstract

Let $f_n$ be a random polynomial of degree $n\ge 2$ whose coefficients are independent and identically distributed random variables. We study the separation distances between roots of $f_n$ and prove that the set of these distances, normalized by $n^{-5/4}$, converges in distribution as $n\to \infty$ to a non-homogeneous Poisson point process. As a corollary, we deduce that the minimal separation distance between roots of $f_n$, normalized by $n^{-5/4}$ has a non-trivial limit law. In the course of the proof, we establish a related result which may be of independent interest: a Taylor series with random i.i.d. coefficients almost-surely does not have a double zero anywhere other than the origin.