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

TL;DR

This paper extends Runge's method for effectively finding integral points on varieties, generalizing it to higher dimensions and coverings, and applies it to solve specific Diophantine equations involving squares in arithmetic progressions.

Contribution

It generalizes Runge's theorem to higher-dimensional varieties and develops new techniques using coverings, including explicit results for superelliptic curves.

Findings

Generalized Runge's theorem for higher-dimensional varieties

Developed methods using coverings to expand Runge's approach

Solved equations involving squares in products of arithmetic progression terms

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