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Definability of Frobenius orbits and a result on rational distance sets

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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).

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Notes

  1. We remark that a similar result is obtained for characteristic 2 replacing “square” by “cube”, provided that we require s odd. For applications in definability, such a version is enough. We leave this straightforward modification to the interested reader.

  2. The reader wishing to extend the proof to the case \(p=2\) might find it convenient to first establish a slightly modified version of Theorem 1.6 concerning cube values rather than square values.

  3. Let us recall that in Vojta’s analogy, what corresponds to a single analytic map to a variety X is an infinite set of rational points of X (as opposed to a single rational point).

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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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Correspondence to Hector Pasten.

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Communicated by U. Zannier.

Supported by a Benjamin Peirce Fellowship at Harvard and by a Schmidt Fellowship and by the NSF at the Institute for Advanced Study. This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Pasten, H. Definability of Frobenius orbits and a result on rational distance sets. Monatsh Math 182, 99–126 (2017). https://doi.org/10.1007/s00605-016-0973-2

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