Heights of Function Field Points on Curves Given by Equations with Separated Variables

Abstract

Let $P$ and $Q$ be polynomials in one variable over an algebraically closed field $k$ of characteristic zero. Let $f$ and $g$ be elements of a function field $\K$ over $k$ such that $P(f)=Q(g).$ We give conditions on $P$ and $Q$ such that the height of $f$ and $g$ can be effectively bounded, and moreover, we give sufficient conditions on $P$ and $Q$ under which $f$ and $g$ must be constant.