Electronic Journal of Probability

Double roots of random polynomials with integer coefficients

Ohad N. Feldheim and Arnab Sen

Full-text: Open access

Abstract

We consider random polynomials whose coefficients are independent and identically distributed on the integers. We prove that if the coefficient distribution has bounded support and its probability to take any particular value is at most $\tfrac 12$, then the probability of the polynomial to have a double root is dominated by the probability that either $0$, $1$, or $-1$ is a double root up to an error of $o(n^{-2})$. We also show that if the support of the coefficients’ distribution excludes $0$, then the double root probability is $O(n^{-2})$. Our result generalizes a similar result of Peled, Sen and Zeitouni [13] for Littlewood polynomials.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 10, 23 pp.

Dates
Received: 8 April 2016
Accepted: 1 January 2017
First available in Project Euclid: 3 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1486090890

Digital Object Identifier
doi:10.1214/17-EJP24

Mathematical Reviews number (MathSciNet)
MR3613703

Zentralblatt MATH identifier
06681512

Subjects
Primary: 60C05: Combinatorial probability 60G50: Sums of independent random variables; random walks

Keywords
random polynomials double roots anti-concentration algebraic numbers

Rights
Creative Commons Attribution 4.0 International License.

Citation

Feldheim, Ohad N.; Sen, Arnab. Double roots of random polynomials with integer coefficients. Electron. J. Probab. 22 (2017), paper no. 10, 23 pp. doi:10.1214/17-EJP24. https://projecteuclid.org/euclid.ejp/1486090890


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