Variations on a theme of Runge: effective determination of integral points on certain varieties

Aaron Levin[1]

  • [1] Centro di Ricerca Matematica Ennio De Giorgi Collegio Puteano Scuola Normale Superiore Piazza dei Cavalieri, 3 I-56100 Pisa, Italy

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 2, page 385-417
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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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 20.2 (2008): 385-417. <http://eudml.org/doc/10843>.

@article{Levin2008,
abstract = {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.},
affiliation = {Centro di Ricerca Matematica Ennio De Giorgi Collegio Puteano Scuola Normale Superiore Piazza dei Cavalieri, 3 I-56100 Pisa, Italy},
author = {Levin, Aaron},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Diophantine equations; integral points},
language = {eng},
number = {2},
pages = {385-417},
publisher = {Université Bordeaux 1},
title = {Variations on a theme of Runge: effective determination of integral points on certain varieties},
url = {http://eudml.org/doc/10843},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Levin, Aaron
TI - Variations on a theme of Runge: effective determination of integral points on certain varieties
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 2
SP - 385
EP - 417
AB - 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.
LA - eng
KW - Diophantine equations; integral points
UR - http://eudml.org/doc/10843
ER -

References

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