Double roots of random Littlewood polynomials

TL;DR

This paper analyzes the probability of double roots in random Littlewood polynomials with coefficients in {-1,1} and extends results to more general coefficient distributions, revealing precise asymptotics based on polynomial degree.

Contribution

It provides exact asymptotic probabilities for double roots in random Littlewood polynomials and generalizes to broader coefficient distributions with detailed asymptotic analysis.

Findings

Probability of double roots is o(n^{-2}) when n+1 not divisible by 4.

Explicit asymptotics for double root probability when n+1 divisible by 4.

Extension of results to polynomials with coefficients supported on {-1, 0, 1}.

Abstract

We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and asymptotic to $\frac{8\sqrt{3}}{\pi n^2}$ otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on { -1, 0, 1} and whose largest atom is strictly less than 1/\sqrt{3}. In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o(n^{-2}) factor and we find the asymptotics of the latter probability.