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

2024/08/26 by Doyle, John R., Hindes, Wade · 3 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.14657

Abstract

Let K be a function field of characteristic p≥0 or a number field over which the abc conjecture holds, and let ϕ(x)=x^d+c ∈ K[x] be a unicritical polynomial of degree d≥2 with d \not≡ 0,1\pmodp. 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(x^d+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.

Cited by

• Preperiodic integers for x^d+c in large degree
• Prime-powered images and irreducible polynomials in dynamical semigroups
• On the factorization of iterates of x^d+c in large degree

Related

• Preperiodic integers for x^d+c in large degree
• Abelian dynamical Galois groups for unicritical polynomials
• Preperiodic points of polynomials over global fields
• The abc-Conjecture implies uniform bounds on dynamical Zsigmondy sets
• Uniform bounds on periodic points of polynomials with good reduction