S-unit equation in two variables and Padé approximations

Abstract

In this paper, we use Padé approximations, constructed in [Kawashima and Poëls, Padé approximations for shifted functions and parametric geometry of numbers, J. Number Theory 243 (2023) 646–687] for binomial functions, to give a new upper bound for the number of the solutions of the [Formula: see text]-unit equation in two variables. Combining explicit Padé approximants with a simple argument relying on Mahler measure as well as the local height, we refine the bound due to Evertse [On equations in [Formula: see text]-units and the Thue-Mahler equation, Invent. Math. 75 (1984) 561–584].