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Vietnam Journal of Mathematics 39:3 (2011) 343-368

Some Remarks on Diophantine Equations and

Diophantine Approximation

Claude Levesque¹ and Michel Waldschmidt²

¹Département de mathématiques et de statistique, Université Laval, Québec

(Québec), Canada G1V 0A6

²Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie

(Paris 6), 4 Place Jussieu, F-75252 Paris Cedex 05, France

Abstract. We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various Diophantine equations. We also give many equivalent statements of Mahler’s generalization of the fundamental theorem of Thue. In particular, we show that the theorem of Thue–Mahler for degree 3 implies the theorem of Thue–Mahler for arbitrary degree $\geq 3$, and we relate it with a theorem of Siegel on the rational integral points of the projective line P¹(K) minus 3 points. Finally we extend our study to higher dimensional spaces in connection with Schmidt’s Subspace Theorem.

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Published by Springer since January 2013