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Double roots of random littlewood polynomials

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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.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Ron Peled

  2. Department of Mathematics, University of Minnesota, 206 Church st. SE, Minneapolis, MN, 55455, USA

    Arnab Sen

  3. Department of Mathematics, The Weizmann Institute of Science POB 26, Rehovot, 76100, Israel

    Ofer Zeitouni

  4. Courant Institute, 251 Mercer Street, New York, NY, 10012, USA

    Ofer Zeitouni

Authors
  1. Ron Peled
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  2. Arnab Sen
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  3. Ofer Zeitouni
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Corresponding author

Correspondence to Ofer Zeitouni.

Additional information

Supported by an ISF grant and an IRG grant.

Supported by NSF grant 1406247.

Supported by an ISF grant and by the Herman P. Taubman professorial chair of Mathematics at WIS.

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Peled, R., Sen, A. & Zeitouni, O. Double roots of random littlewood polynomials. Isr. J. Math. 213, 55–77 (2016). https://doi.org/10.1007/s11856-016-1328-3

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  • DOI: https://doi.org/10.1007/s11856-016-1328-3

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