Vol. 8, No. 3, 2014

Download this article
Download this article For screen
For printing
Recent Issues

Volume 18
Issue 6, 1039–1219
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author index
To appear
Other MSP journals
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