Bornes effectives pour les solutions des équations enS-unités et des équations de Thue–Mahler

Abstract

Recently, Bombieri and Bombieri and Cohen have developped a new method in effective diophantine approximation, based on a reworked version of the Thue principle, the Dyson lemma and some geometry of numbers. In the present work, we follow their approach, utilizing however estimates for linear forms in two or three logarithms instead of the Thue principle. This tool allows us to sharpen their results. We apply our main estimates to provide new explicit—and, in a certain sense, improved—upper bounds for the size of the solutions of the S-unit equation and of the Thue–Mahler equation. We also derive a new effective irrationality measure for algebraic numbers and compare our estimates with the recent results of Bugeaud and Győry.