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
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.
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.
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).
References
Anning, N., Erdös, P.: Integral distances. Bull. Am. Math. Soc. 51, 598–600 (1945)
Cherry, W.: Lecture notes on non-Archimedean function theory, Advanced school on \(p\)-adic analysis and applications, The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy (2009). Preprint, available at arXiv:0909.4509
Dvir, Z.: On the size of Kakeya sets in finite fields. J. Am. Math. Soc. 22, 1093–1097 (2009)
Eisenträger, K., Shlapentokh, A.: Undecidability in function fields of positive characteristic. Int. Math. Res. Not. IMRN 21, 4051–4086 (2009)
Eisenträger, K., Shlapentokh, A.: Hilbert’s tenth problem over function fields of positive characteristic not containing the algebraic closure of a finite field. J. Eur. Math. Soc. (2015) (To appear)
Elkies, N.: ABC implies Mordell. Int. Math. Res. Not. 7, 99–109 (1991)
Garcia-Fritz, N., Pasten, H.: Uniform positive existential interpretation of the integers in rings of entire functions of positive characteristic. J. Number Theory 156, 368–393 (2015)
Hu, P. C., Yang, C.-C.: Value distribution theory related to number theory. Birkhauser Verlag, Basel (2006). ISBN: 978-3-7643-7568-3; 3-7643-7568-X
Lipshitz, L., Pheidas, T.: An analogue of Hilbert’s tenth problem for \(p\)-adic entire functions. J. Symb. Logic 60(4), 1301–1309 (1995)
Makhul, M., Shaffaf, J.: On uniform boundedness of a rational distance set in the plane. C. R. Math. Acad. Sci. Paris (English, French summary) 350(3–4), 121–124 (2012)
Matiyasevich, Y.: Enumerable sets are diophantine. Dokladii Akademii Nauk SSSR 191, 279–282 (1970)
Matiyasevich, Y.: English translation. Sov. Math. Dokl. 11, 354–358 (1970)
Pasten, H.: Representation of squares by monic second degree polynomials in the field of p-adic meromorphic functions. Trans. Am. Math. Soc. 364(1), 417–446 (2012)
Pasten, H.: Powerful values of polynomials and a conjecture of Vojta. J. Number Theory 133(9), 2964–2998 (2013)
Pasten, H., Pheidas, T., Vidaux, X.: Uniform existential interpretation of arithmetic in rings of functions of positive characteristic. Invent. Math. 196(2), 453–484 (2014)
Pasten, H., Wang, J.: Extensions of Büchi’s higher powers problem to positive characteristic. IMRN (2014) (to appear)
Pheidas, T.: Hilbert’s tenth problem for fields of rational functions over finite fields. Invent. Math. 103(1), 1–8 (1991)
Pheidas, T., Vidaux, X.: Corrigendum: the analogue of Büchi’s problem for rational functions. J. Lond. Math. Soc. (2) 82(1), 273–278 (2010)
Pheidas, T., Zahidi, K.: Undecidability of existential theories of rings and fields: a survey. Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), vol. 270, pp. 49–105. Contemp. Math., Amer. Math. Soc., Providence, RI (2000)
Robinson, J.: Definability and decision problems in arithmetic. J. Symb. Logic 14, 98–114 (1949)
Robinson, R.: An essentially undecidable axiom system. In: Proceedings of the International Congress of Mathematicians 1950, vol. 1, pp. 729–730. American Mathematical Society, Providence (1952)
Shaffaf, J.: A proof of the Erd’os–Ulam problem assuming Bombieri–Lang conjecture. Preprint available at arXiv:1501.00159
Shlapentokh, A.: Diophantine relations between rings of S-integers of fields of algebraic functions in one variable over constant fields of positive characteristic. J. Symb. Logic 58(1), 158–192 (1993)
Shlapentokh, A., Vidaux, X.: The analogue of Büchi’s problem for function fields. J. Algebra 330, 482–506 (2011)
Solymosi, J., de Zeeuw, F.: On a question of Erdös and Ulam. Discrete Comput. Geom. 43(2), 393–401 (2010)
Tao, T.: Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory. EMS Surv. Math. Sci. 1(1), 1–46 (2014)
Tao, T.: The Erdös–Ulam problem, varieties of general type, and the Bombieri–Lang conjecture. Post (2014/12/20) in http://terrytao.wordpress.com
Ulam, S.: A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics, vol. 8. Interscience, New York (1960)
Vidaux, X.: An analogue of Hilbert’s 10th problem for fields of meromorphic functions over non-Archimedean valued fields. J. Number Theory 101(1), 48–73 (2003)
Vojta, P.: Diophantine approximation and Nevanlinna theory. In: Arithmetic Geometry, Lecture Notes in Math., vol. 2009, pp. 111–224. Springer, Berlin (2011)
Vojta, P.: Diagonal quadratic forms and Hilbert’s Tenth Problem. Contemp. Math. 270, 261–274 (2000)
Wang, J.: The truncated second main theorem of function fields. J. Number Theory 58(1), 139–157 (1996)
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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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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DOI: https://doi.org/10.1007/s00605-016-0973-2