On the irreducibility of $f(2^n,3^m,X)$ and other such polynomials

Abstract

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