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Definable valuations induced by multiplicative subgroups and NIP fields

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Abstract

We study the algebraic implications of the non-independence property and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a (definable) henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson’s “The canonical topology on dp-minimal fields” (J Math Log 18(2):1850007, 2018).

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References

  1. Bélair, L.: Types dans les corps valués munis d’applications coefficients. Illinois J. Math. 43(2), 410–425 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cherlin, G., Shelah, S.: Superstable fields and groups. Ann. Math. Logic 18(3), 227–270 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chernikov, A.: Theories without the tree property of the second kind. Ann. Pure Appl. Logic 165(2), 695–723 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Delon, F.: Types sur \({\bf C}((X))\). In: Study Group on Stable Theories (Bruno Poizat), Second year: 1978/79 (French), pages Exp. No. 5, 29. Secrétariat Math., Paris (1981)

  5. Dupont, K.: Definable Valuations Induced by Definable Subgroups. Groups, Modules and Model Theory - Surveys and Recent Developments in Memory of Rüdiger Göbel, 83–109 Springer, Berlin (2017)

  6. Ealy, C., Krupiński, K., Pillay, A.: Superrosy dependent groups having finitely satisfiable generics. Ann. Pure Appl. Log. 151(1), 1–21 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Efrat, I.: Valuations, orderings, and Milnor \(K\)-theory. Mathematical Surveys and Monographs, vol. 124. American Mathematical Society, Providence (2006)

  8. Engler, A.J., Prestel, A.: Valued Fields. Springer Monographs in Mathematics. Springer, Berlin (2005)

    MATH  Google Scholar 

  9. Gurevich, Y., Schmitt, P.H.: The theory of ordered abelian groups does not have the independence property. Trans. Am. Math. Soc. 284(1), 171–182 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Halevi, Y., Hasson, A.: Strongly dependent ordered abelian groups and henselian fields. Israel J. Math. (to appear), available at https://arxiv.org/abs/1706.03376 (2017)

  11. Halevi, Y., Hasson, A., Janhke, F.: A conjectural classification of strongly dependent fields. Bull. of Symb. Log. (to appear), available at https://arxiv.org/pdf/1805.03814.pdf (2018)

  12. Halevi, Y., Hasson, A.: Eliminating field quantifiers in strongly dependent fields. In: Proceedings of AMS (to appear) Available at http://de.arxiv.org/abs/1707.03188 (2017)

  13. Hong, J.: Definable non-divisible Henselian valuations. Bull. Lond. Math. Soc. 46(1), 14–18 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hrushovski, E., Peterzil, Y., Pillay, A.: Groups, measures, and the NIP. J. Am. Math. Soc. 21(2), 563–596 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jahnke, F., Koenigsmann, J.: Definable henselian valuations. J. Symb. Log. 80(1), 85–99 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jahnke, F., Koenigsmann, J.: Uniformly defining \(p\)-henselian valuations. Ann. Pure Appl. Log. 166(7–8), 741–754 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jahnke, F., Simon, P.: NIP henselian valued fields. ArXiv e-prints (2016)

  18. Jahnke, F., Simon, P., Walsberg, E.: Dp-Minimal Valued Fields. To appear in J. Symb. Log.

  19. Johnson, W.: The canonical topology on dp-minimal fields. J. Math. Log. 18(2), 1850007 (2018)

  20. Johnson, W.: On dp-minimal fields. arXiv:1507.02745, http://front.math.ucdavis.edu/1507.02745 (2015)

  21. Kaplan, I., Scanlon, T., Wagner, F.O.: Artin-Schreier extensions in NIP and simple fields. Israel J. Math. 185, 141–153 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kaplan, I., Shelah, S.: Chain conditions in dependent groups. Ann. Pure Appl. Log. 164(12), 1322–1337 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Koenigsmann, J.: \(p\)-henselian fields. Manuscripta Math. 87(1), 89–99 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Koenigsmann, J.: Definable valuations. Prèpublications de L’Equipe de Logique, 54 (1995)

  25. Krupiński, K.: Fields interpretable in rosy theories. Israel J. Math. 175, 421–444 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Krupiński, K.: Superrosy fields and valuations. Ann. Pure Appl. Log. 166(3), 342–357 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pillay, A., Scanlon, T., Wagner, F.O.: Supersimple fields and division rings. Math. Res. Lett. 5(4), 473–483 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pillay, A., Poizat, B.: Corps et chirurgie. J. Symbol. Log. 60(2), 528–533 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Poizat, B.: Stable groups, vol. 87 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI 2001. Translated from the 1987 French original by Moses Gabriel Klein

  30. Shelah, S.: Classification theory for elementary classes with the dependence property—a modest beginning. Sci. Math. Jpn. 59(2), 265–316 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Shelah, S.: Strongly dependent theories. Israel J. Math. 204(1), 1–83 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wagner, F.O.: Simple Theories. Mathematics and its Applications, vol. 503. Kluwer, Dordrecht (2000)

    Book  Google Scholar 

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Acknowledgements

We would like to thank I. Efrat, M. Hils, F. Jahnke, M. Kamensky, F.-V. Kuhlmann and P. Simon for several ideas, corrections and suggestions.

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Authors and Affiliations

  1. FB Mathematik und Statistik, Universität Konstanz, 78457, Constance, Germany

    Katharina Dupont & Salma Kuhlmann

  2. Department of mathematics, Ben Gurion University of the Negev, Beersheba, Israel

    Assaf Hasson

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  1. Katharina Dupont
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  2. Assaf Hasson
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  3. Salma Kuhlmann
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Correspondence to Assaf Hasson.

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K. Dupont: Partially supported by a Minerva fellowship and by ISF grant No. 23/09.

A. Hasson: Partially supported by GIF Grant 2165/2011 and by ISF Grant 181/16.

S. Kuhlmann: Partially supported by AFF grant from the University of Konstanz.

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Dupont, K., Hasson, A. & Kuhlmann, S. Definable valuations induced by multiplicative subgroups and NIP fields. Arch. Math. Logic 58, 819–839 (2019). https://doi.org/10.1007/s00153-019-00661-2

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