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2014 Linear forms in logarithms and integral points on higher-dimensional varieties
Aaron Levin
Algebra Number Theory 8(3): 647-687 (2014). 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|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.

Citation

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Aaron Levin. "Linear forms in logarithms and integral points on higher-dimensional varieties." Algebra Number Theory 8 (3) 647 - 687, 2014. https://doi.org/10.2140/ant.2014.8.647

Information

Received: 14 April 2013; Revised: 2 September 2013; Accepted: 2 October 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1371.11107
MathSciNet: MR3218805
Digital Object Identifier: 10.2140/ant.2014.8.647

Subjects:
Primary: 11G35
Secondary: 11D61 , 11J86

Keywords: Integral points , linear forms in logarithms , Runge's method , unit equation

Rights: Copyright © 2014 Mathematical Sciences Publishers

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Vol.8 • No. 3 • 2014
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