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

Let K be a function field or a number field over which the $abc$ conjecture holds, and let φ(x)=xᵈ+c ∈ K[x] be a unicritical polynomial of degree d≥2 not divisible by the characteristic of K. We completely classify all portraits of K-rational preperiodic points for such φ for all sufficiently large degrees d. More precisely, we prove that, up to accounting for the natural action of dth roots of unity on the preperiodic points for φ, 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 φ is uniformly bounded, independent of d. That is, there is a constant $B(K)$ depending only on K such that |PrePer(xᵈ+c,K)|≤ B(K) for all d≥2 and all c∈ 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.