p-adic uniformization and the action of Galois on certain affine correspondences

TL;DR

This paper studies the Galois action on iterated preimages of certain polynomial correspondences over number fields, establishing a p-adic uniformization that reveals finiteness properties of Galois orbits.

Contribution

It introduces a p-adic uniformization method for affine correspondences, leading to new finiteness results on Galois orbits of preimages in polynomial dynamical systems.

Findings

Finitely many Galois orbits of preimages under specified conditions

Established p-adic uniformization for polynomial correspondences

Connected p-adic uniformization to Galois orbit finiteness

Abstract

Given two monic polynomials f and g with coefficients in a number field K, and some a in K, we examine the action of the absolute Galois group of K on the directed graph of iterated preimages of a under the correspondence g(y)=f(x), assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of K exists at which f and g have integral coefficients, and at which a is not integral, we show that this directed graph of preimages consists of finitely many Galois-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Bottcher's uniformization of polynomial dynamical systems over the complex numbers.