Variations on a theme of Runge: effective determination of integral points on certain varieties
Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 385-417.

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge’s theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge’s theorem due to Bombieri. We then take up the study of how Runge’s method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our method, we completely solve certain equations involving squares in products of terms in an arithmetic progression.

Nous considérons quelques variations sur la méthode classique de Runge pour déterminer effectivement les points entiers sur certaines courbes. Nous prouvons d’abord une version du théorème de Runge valide pour des variétés de dimension supérieure, généralisant une version uniforme du théorème de Runge due à Bombieri. Nous étudions alors comment la méthode de Runge peut être étendue en utilisant certains revêtements. Nous prouvons un résultat pour les courbes arbitraires et un résultat plus explicite pour les courbes superelliptic. Comme application de notre méthode, nous résolvons complètement certaines équations impliquant des carrés dans les produits des termes dans une progression arithmétique.

DOI: 10.5802/jtnb.634
Levin, Aaron 1

1 Centro di Ricerca Matematica Ennio De Giorgi Collegio Puteano Scuola Normale Superiore Piazza dei Cavalieri, 3 I-56100 Pisa, Italy
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Levin, Aaron. Variations on a theme of Runge: effective determination of integral points on certain varieties. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 385-417. doi : 10.5802/jtnb.634. http://www.numdam.org/articles/10.5802/jtnb.634/

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