Irreducibility of iterates of post-critically finite quadratic polynomials over $\mathbb{Q}$

TL;DR

This paper classifies monic, post-critically finite quadratic polynomials over  with iterates reducible modulo every prime but irreducible over , providing new examples and addressing questions on irreducibility stability.

Contribution

It offers a near-complete classification of such polynomials, identifies all integers with irreducible iterates for a specific polynomial family, and proposes a conjecture on stability criteria.

Findings

Classified all such quadratic polynomials with few exceptions.

Identified all integers making iterates of a specific polynomial irreducible.

Proposed a conjecture on the stability condition for these polynomials.

Abstract

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over $\mathbb{Q}$. In particular, we obtain infinitely many new examples of the phenomenon studied in \cite{Jones}. While doing this, we also find, up to three possible exceptions, all integers $a$ such that all iterates of the quadratic polynomial $(x+a)^2-a-1$ are irreducible over $\mathbb{Q}$, which answers a question posed in \cite{AyadMcdonald}, except for three values of $a$. Finally, we make a conjecture that suggests a necessary and sufficient condition for the stability of any monic, post-critically finite quadratic polynomial over any field of characteristic $\neq 2$.