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

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 $f\left(u,v\right)=w$, where $u$, $v$, and $w$ are $S$-units, $\left|S\right|\le 3$, 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.