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

TL;DR

This paper investigates the irreducibility of specialized polynomials with exponential parameters, establishing conditions under which they remain irreducible for most integer exponents, assuming certain ramification conditions and the GRH.

Contribution

It provides new criteria for the irreducibility of exponential specializations of multivariate polynomials, extending previous results under specific ramification and GRH assumptions.

Findings

Most such specialized polynomials are irreducible for almost all integer exponents.

Conditional on GRH, the irreducibility holds under a ramification assumption.

The results apply to polynomials with exponential parameters outside b1 1.

Abstract

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 $\mathbb{Q}[X]$.