Arboreal Galois representations and uniformization of polynomial dynamics

TL;DR

This paper constructs a Galois-equivariant biholomorphic change of variables near infinity for polynomial dynamics over local fields, linking Galois theory of preimages to multiplicative Kummer theory and generalizing complex uniformization concepts.

Contribution

It introduces a new biholomorphic transformation that simplifies polynomial dynamics over local fields and connects Galois theory with multiplicative uniformization techniques.

Findings

Biholomorphic change of variables transforms polynomial action to multiplicative action.

The construction is Galois-equivariant, enabling algebraic insights.

Links polynomial preimage Galois groups to Kummer theory.

Abstract

Given a polynomial f of degree d defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z->f(z) to the multiplicative action z->z^d. The relation between this construction and the Bottcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves, and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory.