Linear forms in logarithms and integral points on higher-dimensional varieties

TL;DR

This paper uses inequalities from linear forms in logarithms to derive explicit bounds on S-integral points on higher-dimensional varieties, extending previous results on curves and solving polynomial unit equations effectively.

Contribution

It introduces a higher-dimensional approach to effective bounds on S-integral points, generalizing known results for curves and providing explicit solutions for polynomial unit equations.

Findings

Explicit bounds for S-integral points on certain affine varieties.

Effective solutions for polynomial unit equations with small S.

Comparison with higher-dimensional Runge's method.

Abstract

We apply inequalities from the theory of linear forms in logarithms to deduce effective results on S-integral points on certain higher-dimensional varieties when the cardinality of S is sufficiently small. These results may be viewed as a higher-dimensional version of an effective result of Bilu on integral points on curves. In particular, we prove a completely explicit result for integral points on certain affine subsets of the projective plane. As an application, we generalize an effective result of Vojta on the three-variable unit equation by giving an effective solution of the polynomial unit equation f(u,v)=w, where u,v, and w are S-units, |S|\leq 3, and f is a polynomial satisfying certain conditions (which are generically satisfied). Finally, we compare our results to a higher-dimensional version of Runge's method, which has some characteristics in common with the results here.