Newly reducible polynomial iterates

Peter Illig, Rafe Jones, Eli Orvis, Yukihiko Segawa, Nick Spinale

[pdf] [arXiv:2008.01222] [doi coming soon]

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.