Baker’s method, relying on estimates on linear forms in logarithms of algebraic
numbers, allows one to prove in several situations the effective finiteness of integral
points on varieties. In this article, we generalize results of Levin regarding Baker’s
method for varieties, and explain how, quite surprisingly, it mixes (under additional
hypotheses) with Runge’s method to improve some known estimates in the case of
curves by bypassing (or more generally reducing) the need for linear forms in
-adic
logarithms. We then use these ideas to improve known estimates on solutions of
-unit
equations. Finally, we explain how a finer analysis and formalism can improve upon
the conditions given, and give some applications to the Siegel modular variety
.
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