Unit equations and Fermat surfaces in positive characteristic

TL;DR

This paper investigates solutions to the three-variable unit equation in positive characteristic function fields, providing bounds on solutions and applying these methods to Fermat surfaces.

Contribution

It introduces new bounds for solutions of the unit equation and applies these results to analyze Fermat surfaces in positive characteristic.

Findings

Upper bounds for the height of solutions

Bounds on the number of solutions

Application to Fermat surfaces

Abstract

In this article we study the three-variable unit equation $x + y + z = 1$ to be solved in $x, y, z \in \mathcal{O}_S^\ast$, where $\mathcal{O}_S^\ast$ is the $S$-unit group of some global function field. We give upper bounds for the height of solutions and the number of solutions. We also apply these techniques to study the Fermat surface $x^N + y^N + z^N = 1$.