Year: 2014
Type: article
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.
Source: Algebra & Number Theory
Author: Aaron Levin
Institution: Michigan State University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Funder: National Science Foundation
OpenAlex ID: W1979974480