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

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

2008-05-09

arxiv.org/abs/0805.1345v1

Mathematics

Number Theory

30 pages

Scientific paper

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 solve certain equations involving squares in products of terms in an arithmetic progression.

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