Backward Orbit Conjecture for the Powering Map over Global Fields

TL;DR

This paper proves the backward orbit conjecture for the powering map over certain function fields, specifically when the degree is coprime to the characteristic, advancing understanding in arithmetic dynamics.

Contribution

It establishes the conjecture's validity for powering maps over function fields with finite constant fields under specific conditions.

Findings

Backward orbit conjecture holds for $$-powering maps over these fields.

The proof applies when $d$ is coprime to the characteristic of the field.

Results contribute to the theory of arithmetic dynamics over function fields.

Abstract

We show that the backward orbit conjecture is true for powering map $\phi(z)=z^d$ over a function field $K$ with a finite field of constants, and when $d$ is relatively prime to the characteristic of $K$.