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

Levin, Aaron

doi:10.2140/ant.2014.8.647

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

Aaron Levin

Vol. 8 (2014), No. 3, 647–687

DOI: 10.2140/ant.2014.8.647

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.

Keywords

integral points, unit equation, linear forms in logarithms, Runge's method

Mathematical Subject Classification 2010

Primary: 11G35

Secondary: 11J86, 11D61

Milestones

Received: 14 April 2013

Revised: 2 September 2013

Accepted: 2 October 2013

Published: 31 May 2014

Authors

Aaron Levin
Department of Mathematics Michigan State University 619 Red Cedar Road East Lansing, MI 48824 United States

Vol. 8, No. 3, 2014