  We classify the polynomials with integral coefficients that, when evaluated
on a group element of finite order $n$, define a unit in the integral group
ring for infinitely many positive integers $n$. We show that this happens if
and only if the polynomial defines generic units in the sense of Marciniak and
Sehgal. We also classify the polynomials with integral coefficients which
provides units when evaluated on $n$-roots of a fixed integer $a$ for
infinitely many positive integers $n$.
