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 $d\equiv 2\mathrm{mod}4$. 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.

arXiv comment18 pages