Vol. 8, No. 3, 2014

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

Aaron Levin

Vol. 8 (2014), No. 3, 647–687
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, $|S|\le 3$, 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.

Keywords
integral points, unit equation, linear forms in logarithms, Runge's method
Mathematical Subject Classification 2010
Primary: 11G35
Secondary: 11J86, 11D61