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.