Mahler's work on Diophantine equations and subsequent developments

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.

26 pages. This paper will appear in "Mahler Selecta", a volume dedicated to the work of Kurt Mahler and its impact