Newly reducible polynomial iterates

Abstract

Given a field $K$ and $n > 1$, we say that a polynomial $f \in K[x]$ has newly reducible $n$th iterate over $K$ if $f^{n-1}$ is irreducible over $K$, but $f^n$ is not (here $f^i$ denotes the $i$th iterate of $f$). We pose the problem of characterizing, for given $d,n > 1$, fields $K$ such that there exists $f \in K[x]$ of degree $d$ with newly reducible $n$th iterate, and the similar problem for fields admitting infinitely many such $f$. We give results in the cases $(d,n) \in \{(2,2), (2,3), (3,2), (4,2)\}$ as well as for $(d,2)$ when $d \equiv 2 \bmod{4}$. In particular, we show that for all these $(d,n)$ pairs, there are infinitely many monic $f \in \mathbb{Z}[x]$ of degree $d$ with newly reducible $n$th iterate over $\mathbb{Q}$. Curiously, the minimal polynomial $x^2 - x - 1$ of the golden ratio is one example of $f \in \mathbb{Z}[x]$ with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.