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Monatshefte für Mathematik

, Volume 182, Issue 1, pp 99–126 | Cite as

Definability of Frobenius orbits and a result on rational distance sets

  • Hector PastenEmail author
Article
  • 165 Downloads

Abstract

We prove that the first order theory of (possibly transcendental) meromorphic functions of positive characteristic \(p>2\) is undecidable. We also establish a negative solution to an analogue of Hilbert’s tenth problem for such fields of meromorphic functions, for Diophantine equations including vanishing conditions. These undecidability results are proved by showing that the binary relation \(\exists s\ge 0, f=g^{p^s}\) is positive existentially definable in such fields. We also prove that the abc conjecture implies a solution to the Erdös–Ulam problem on rational distance sets. These two seemingly distant topics are addressed by a study of power values of bivariate polynomials of the form F(X)G(Y).

Keywords

Undecidability Positive characteristic Abc conjecture Erdös–Ulam problem Rational distance sets 

Mathematics Subject Classification

Primary 11U05 Secondary 11J97 52C10 

Notes

Acknowledgments

I thank Alexandra Shlapentokh for some useful references and Julie T.-Y. Wang for comments and corrections on an earlier version of this manuscript. I am indebted to Xavier Vidaux for several suggestions that improved the presentation of this paper, and to Pierre Deligne for asking some questions that led me to include Theorem 1.3 and other improvements. Also, I heartily thank the referee for providing valuable feedback and corrections. This work originated as an attempt to present a simplified proof of Pheidas theorem from [17] in the graduate course Diophantine Definability (Math 259) taught at Harvard during the Spring term of 2015. I thank the Mathematics Department at Harvard for giving me the opportunity to teach this course.

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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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