On effective $\epsilon$-integrality in orbits of rational maps over function fields and multiplicative dependence

TL;DR

This paper establishes effective bounds for quasi-integral points in orbits of non-isotrivial rational maps over function fields and proves finiteness results for algebraic functions with multiplicatively dependent orbit elements.

Contribution

It generalizes previous results by providing effective bounds and finiteness theorems over function fields, extending work from characteristic zero to broader settings.

Findings

Effective bounds for quasi-integral points in orbits

Finiteness results for algebraic functions with multiplicative dependence

Generalization of previous work to broader function field contexts

Abstract

We give effective bounds for the set quasi-integral points in orbits of non-isotrivial rational maps over function fields under some conditions, generalizing previous work of Hsia and Silverman (2011) for orbits over function fields of characteristic zero. We then use this to prove finiteness results for algebraic functions whose orbit under a rational function has multiplicative dependent elements modulo rings of $S$-integers, generalizing recent results over number fields.