Local fields, iterated extensions, and Julia Sets

TL;DR

This paper classifies the extension behavior of iterated preimages under unicritical polynomials over complete fields, showing it depends solely on the valuation of the parameter and connecting ramification to Berkovich Julia sets.

Contribution

It removes previous restrictions on the degree of the polynomial, completing the classification for all degrees and relating ramification to Berkovich Julia set structure.

Findings

Extension behavior depends only on valuation of c

Classification now complete for all degrees ≥ 2

Ramification relates to Berkovich Julia set structure

Abstract

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. For $\alpha \in K$, let $K_\infty$ be the extension obtained by adjoining all iterated preimages of $\alpha$ under a unicritical polynomial $f_c(z)=z^\ell - c \in K[z]$. We study the extension $K_\infty/K$ and show that its qualitative behavior depends only on the valuation of $c$. This removes the previous restrictions on $\ell$ in work of Anderson--Hamblen--Poonen--Walton and completes the classification for all $\ell \ge 2$. We also relate the ramification to the structure of the Berkovich Julia set of $f_c$.