TY - JOUR

T1 - Double roots of random littlewood polynomials

AU - Peled, Ron

AU - Sen, Arnab

AU - Zeitouni, Ofer

N1 - Publisher Copyright:
© 2016, Hebrew University of Jerusalem.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s11856-016-1328-3

DO - 10.1007/s11856-016-1328-3

M3 - Article

AN - SCOPUS:84975789326

VL - 213

SP - 55

EP - 77

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -