Double roots of random polynomials with integer coefficients

TL;DR

This paper investigates the probability of double roots in random integer-coefficient polynomials, showing it is primarily influenced by roots at 0, 1, or -1, and providing bounds based on the coefficient distribution.

Contribution

It extends previous results by characterizing the double root probability for a broad class of integer-coefficient polynomials, including bounds and dominant roots.

Findings

Double roots mainly occur at 0, 1, or -1.

Probability of double roots at these points is dominant.

Bounds are established for distributions excluding zero.

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 $\tfrac12$, 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 coefficient distribution excludes $0$ then the double root probability is $O(n^{-2})$. Our result generalizes a similar result of Peled, Sen and Zeitouni for Littlewood polynomials.