Mahler's work on Diophantine equations and subsequent developments

TL;DR

This paper reviews Mahler's foundational work on Diophantine approximation and explores subsequent advances in solving Diophantine equations, including Thue-Mahler and S-unit equations, emphasizing effective finiteness and solution bounds.

Contribution

It synthesizes Mahler's original contributions with recent developments in Diophantine equations, highlighting new bounds and effective methods for finiteness results.

Findings

Estimates for p-adic logarithmic forms improve solution bounds.

Effective finiteness results for Thue-Mahler equations are established.

Connections between Mahler's work and higher-dimensional decomposable form equations are discussed.

Abstract

We discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, S-unit equations and S-integral points on elliptic curves, and go into later developments concerning the number of solutions to Thue-Mahler equations and effective finiteness results for Thue-Mahler equations. For the latter we need estimates for p-adic logarithmic forms, which may be viewed as an outgrowth of Mahler's work on the p-adic Gel'fond-Schneider theorem. We also go briefly into decomposable form equations, these are certain higher dimensional generalizations of Thue-Mahler equations.