Abelian dynamical Galois groups for unicritical polynomials

Andrews and Petsche proposed in 2020 a conjectural characterization of all pairs $(f,\alpha)$, where $f$ is a polynomial over a number field $K$ and $\alpha\in K$, such that the dynamical Galois group of the pair $(f,\alpha)$ is abelian. In this paper we focus on the case of unicritical polynomials $f$, and more general dynamical systems attached to sequences of unicritical polynomials. After having reduced the conjecture to the post-critically finite case, we establish it for all polynomials with periodic critical orbit, over any number field. We next establish the conjecture in full for all monic unicritical polynomials over any quadratic number field. Finally we show that for any given degree $d$ there exists a finite, explicit set of unicritical polynomials that depends only on $d$, such that if $f=ux^d+1$ is a unicritical polynomial over a number field $K$ that lies outside such exceptional set, then there are at most finitely many basepoints $\alpha$ such that the dynamical Galois group of $(f,\alpha)$ is abelian. To obtain these results, we exploit in multiple ways the group theory of the generic dynamical Galois group to force diophantine relations in dynamical quantities attached to $f$. These relations force in all cases, outside of the ones conjectured by Andrews--Petsche, a contradiction either with lower bounds on the heights in abelian extensions, in the style of Amoroso--Zannier, or with the computation of rational points on explicit curves, carried out with techniques from Balakrishnan--Tuitman and Siksek.