Open Access
2017 Double roots of random polynomials with integer coefficients
Ohad N. Feldheim, Arnab Sen
Electron. J. Probab. 22: 1-23 (2017). DOI: 10.1214/17-EJP24


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.


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Ohad N. Feldheim. Arnab Sen. "Double roots of random polynomials with integer coefficients." Electron. J. Probab. 22 1 - 23, 2017.


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

zbMATH: 06681512
MathSciNet: MR3613703
Digital Object Identifier: 10.1214/17-EJP24

Primary: 60C05 , 60G50

Keywords: algebraic numbers , anti-concentration , double roots , random polynomials

Vol.22 • 2017
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