  Let $f(t_1, \ldots, t_r, X)\in \mathbb{Z}[t_1, \ldots, t_r,X]$ be irreducible
and let $a_1, \ldots, a_r\in \mathbb{Z} \smallsetminus \{0,\pm 1\}$. Under a
necessary ramification assumption on $f$, and conditionally on the Generalized
Riemann Hypothesis, we show that for almost all integers $n_1, \ldots, n_r$,
the polynomial $f(a_1^{n_1}, \ldots, a_r^{n_r}, X)$ is irreducible in
$\mathbb{Q}[X]$.
