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

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| 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.

integral points, unit equation, linear forms in logarithms, Runge's method
Mathematical Subject Classification 2010
Primary: 11G35
Secondary: 11J86, 11D61
Received: 14 April 2013
Revised: 2 September 2013
Accepted: 2 October 2013
Published: 31 May 2014
Aaron Levin
Department of Mathematics
Michigan State University
619 Red Cedar Road
East Lansing, MI 48824
United States