$S$-unit equation in two variables and Pad\'{e} approximations

TL;DR

This paper employs Padé approximations of binomial functions to derive a new upper bound on the solutions of the $S$-unit equation, refining previous bounds with explicit formulas and height arguments.

Contribution

It introduces a novel approach using Padé approximations to improve bounds on $S$-unit solutions, combining explicit formulas with Mahler measure and local height techniques.

Findings

New upper bound for $S$-unit solutions

Refinement of Evertse's bound

Explicit formulas for Padé approximants

Abstract

In this article, we use Pad\'{e} approximations constructed for binomial functions, to give a new upper bound for the number of the solutions of the $S$-unit equation. Combining explicit formulae of these Pad\'{e} approximants with a simple argument relying on Mahler measure and on the local height, we refine the bound due to J.-H. Evertse.