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Inventiones mathematicae

, Volume 215, Issue 2, pp 493–533 | Cite as

Greatest common divisors and Vojta’s conjecture for blowups of algebraic tori

  • Aaron LevinEmail author
Article
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Abstract

We give results and inequalities bounding the greatest common divisor of multivariable polynomials evaluated at S-unit arguments, generalizing to an arbitrary number of variables results of Bugeaud–Corvaja–Zannier, Hernández–Luca, and Corvaja–Zannier. In closely related results, and in line with observations of Silverman, we prove special cases of Vojta’s conjecture for blowups of toric varieties. As an application, we classify when terms from simple linear recurrence sequences can have a large greatest common divisor (in an appropriate sense). The primary tool used in the proofs is Schmidt’s Subspace Theorem from Diophantine approximation.

Mathematics Subject Classification

Primary 11J25 Secondary 11G35 11B37 

Notes

Acknowledgements

The author would like to thank Joe Silverman for comments on an earlier draft of the paper. The author would like to thank the referees for the many outstanding comments which improved the quality of the paper, and especially for remarks and observations outlining and leading to a simplified proof of Theorem 1.1.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

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