## Electronic Journal of Probability

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

### Double roots of random polynomials with integer coefficients

Ohad N. Feldheim and Arnab Sen

#### 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