Double roots of random littlewood polynomials

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 \(1/\sqrt 3 \) 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 \(\frac{{8\sqrt 3 }}{{\pi {n^2}}}\). 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.