Unicritical polynomials over $abc$-fields: from uniform boundedness to dynamical Galois groups

TL;DR

This paper classifies all possible preperiodic point structures for unicritical polynomials over certain fields, proving a uniform bound on the number of rational preperiodic points and exploring implications for dynamical Galois groups.

Contribution

It provides a complete classification of preperiodic portraits for unicritical polynomials over fields satisfying the $abc$ conjecture, establishing uniform bounds and applications to Galois groups.

Findings

Exactly thirteen preperiodic portraits up to isomorphism for large degrees

Uniform bound on the number of rational preperiodic points independent of degree

Construction of irreducible polynomials with large dynamical Galois groups

Abstract

Let $K$ be a function field of characteristic $p\geq0$ or a number field over which the $abc$ conjecture holds, and let $\phi(x)=x^d+c \in K[x]$ be a unicritical polynomial of degree $d\geq2$ with $d \not\equiv 0,1\pmod{p}$. We completely classify all portraits of $K$-rational preperiodic points for such $\phi$ for all sufficiently large degrees $d$. More precisely, we prove that, up to accounting for the natural action of $d$th roots of unity on the preperiodic points for $\phi$, there are exactly thirteen such portraits up to isomorphism. In particular, for all such global fields $K$, it follows from our results together with earlier work of Doyle-Poonen and Looper that the number of $K$-rational preperiodic points for $\phi$ is uniformly bounded -- independent of $d$. That is, there is a constant $B(K)$ depending only on $K$ such that \[\big|\text{PrePer}(x^d+c,K)\big|\leq B(K)\] for all $d\geq2$ and all $c\in K$. Moreover, we apply this work to construct many irreducible polynomials with large dynamical Galois groups in semigroups generated by sets of unicritical polynomials under composition.