Double roots of random littlewood polynomials

Ron Peled, Arnab Sen, Ofer Zeitouni

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)55-77
Number of pages23
JournalIsrael Journal of Mathematics
Volume213
Issue number1
DOIs
StatePublished - Jun 1 2016

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