Reducibility of $r{x}^m+p^{e}f(x)$ for Large Primes $p$

Abstract

We study the reducibility over $\mathbb{Q}$ of $F_{p,e}(x)=r{x}^m+p^{e}f(x)$, where $r\in \mathbb{Z}\setminus \{0\}$, $f\in \mathbb{Z}[x]$, and $0\le m<n:=\mathrm{deg}f$, for primes $p$ above explicit coefficient-dependent thresholds. For arbitrary $e\ge 1$, we determine the degrees, endpoint $p$-adic valuations, reductions modulo $p$, and heights of all nonconstant proper integral factors. Writing $\delta =\gcd (e,m,n)$, we show that every such factor has degree $tn/\delta$ for some $1\le t\le \delta -1$; in particular, $\delta =1$ implies irreducibility. For $e=2$ and $e=3$, we obtain effective necessary and sufficient criteria in terms of at most two associated binary quadratic forms and at most one associated binary cubic form, respectively. The quadratic criterion reduces to at most two fixed Pell-type equations with prescribed second coordinate $p$. In the cubic case, Thue's theorem yields finiteness of the reducible primes whenever the associated form is irreducible over $\mathbb{Q}$.