  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.
