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Definability of types and VC density in differential topological fields

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Abstract

Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.

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References

  1. Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., Starchenko, S.: Vapnik–Chervonenkis density in some theories without the independence property II. Notre Dame J. Form. Logic 54(3–4), 311–363 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., Starchenko, S.: Vapnik–Chervonenkis density in some theories without the independence property I. Trans. Am. Math. Soc. 368(8), 5889–5949 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bélair, L.: Le théorème de Macintyre, un théorème de Chevalley p-adique. Ann. Sci. Math. Québec 14(2), 109–120 (1990)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. Brouette, Q., Cousins, G., Pillay, A., Point, F.: Embedded Picard–Vessiot extensions. arXiv:1709.00046 (submitted)

  6. Brouette, Q., Cubides Kovacsics, P., Point, F.: Strong density of definable types and closed ordered differential fields. arXiv:1704.08396 (submitted)

  7. Brouette, Q., Point F.: On differential Galois groups of strongly normal extensions. arXiv:1512.05998 (to appear in Mathematical Logic Quarterly)

  8. Brouette, Q.: Differential algebra, ordered fields and model theory. Ph.D. Thesis UMons (2015)

  9. Brouette, Q.: Definable types in the theory of closed ordered differential fields. Arch. Math. Logic 56(1), 119–129 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. Chernikov, A., Kaplan, I., Simon, P.: Groups and fields with NTP2. Proc. Am. Math. Soc. 143(1), 395–406 (2015)

    Article  MATH  Google Scholar 

  12. Cubides Kovacsics, P., Delon, F.: Definable types in algebraically closed valued fields. Math. Logic Quart. 62(1–2), 35–45 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duret, J.-L.: Les corps faiblement algébriquement clos non séparablement clos ont la propriété d’indépendence [Nonseparably closed weakly algebraically closed fields have the independence property]. In: Model Theory of Algebra and Arithmetic (Proc. Conf., Karpacz, 1979). Lecture Notes in Mathematics, vol. 834, pp. 136–162. Springer, Berlin (1980)

  14. Guingona, V.: On uniform definability of types over finite sets. J. Symb. Logic 77(2), 499–514 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guzy, N., Point, F.: Topological differential fields. Ann. Pure Appl. Logic 161(4), 570–598 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guzy, N., Point, F.: Topological differential fields and dimension functions. J. Symb. Logic 77(4), 1147–1164 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hrushovski, E.: Imaginaries and definable types in algebraically closed valued fields. In: Valuation Theory in Interaction, EMS Series of Congress Report, European Mathematical Society, Zürich, pp. 279–319 (2014)

  18. Jarden, M.: On the model companion of the theory of e-fold ordered fields. Acta Math. 150(3–4), 243–253 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Johnson W.: Fun with fields. Ph.D. thesis, University of California, Berkeley (2016)

  20. Johnson, J.: Differential dimension polynomials and a fundamental theorem on differential modules. Am. J. Math. 91, 239–248 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kaplansky, I.: Maximal fields with valuations. Duke Math. J. 9, 303–321 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marker, D.: Model theory of differential fields. In: Model Theory of Fields (Chapter 2), Lecture Notes in Logic, vol. 5, pp. 38–113. Springer, New York (1996)

  23. Marker, D., Steinhorn, C.: Definable types in o-minimal theories. J. Symb. Logic 59(1), 185–198 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mathews, L.: Cell decomposition and dimension functions in first-order topological structures. Proc. Lond. Math. Soc. 70(1), 1–32 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mellor, T.: Imaginaries in real closed valued fields. Ann. Pure Appl. Logic 139(1–3), 230–279 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Michaux, C., Rivière, C.: Quelques remarques concernant la théorie des corps ordonnés différentiellement clos. Bull. Belg. Math. Soc. Simon Stevin 12(3), 341–348 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Montenegro, S.: Théorie des modèles des corps pseudo-réels clos et pseudo-p-adiquement clos, Thèse de doctorat. Université Paris-Diderot (2015)

  28. Montenegro, S.: Pseudo real closed fields, pseudo p-adically closed fields and NTP2. Ann. Pure Appl. Logic 168(1), 191–232 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pillay, A.: Definability of types, and pairs of o-minimal structures. J. Symb. Logic 59(4), 1400–1409 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Point, F.: Definable groups in topological differential fields. ArXiv:1707.07881

  31. Point, F.: Ensembles définissables dans les corps ordonnés différentiellement clos (On differentially closed ordered fields). C. R. Acad. Sci. Paris Ser. I 349, 929–933 (2011)

    Article  MATH  Google Scholar 

  32. Pong, W.Y.: Some applications of ordinal dimensions to the theory of differentially closed fields. J. Symb. Logic 65(1), 347–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pop, F.: Embedding problems over large fields. Ann. Math. 144(1), 1–34 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Prestel, A.: Pseudo real closed fields. In: Set Theory and Model Theory (1979), Lecture Notes in Mathematics, vol. 872, pp. 127–156. Springer, Berlin (1981)

  35. Singer, M.: The model theory of ordered differential fields. J. Symb. Logic 43(1), 82–91 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  36. Solanki, N.: Uniform companions for expansions of large differential fields. Ph.D. thesis, University of Manchester (2014)

  37. Tressl, M.: The uniform companion for large differential fields of characteristic \(0\). Trans. Am. Math. Soc. 357(10), 3933–3951 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. van den Dries, L.: Model theory of fields, Thesis (Utrecht) (1978)

  39. van den Dries, L.: Algebraic theories with Definable Skolem functions. J. Symb. Logic 49(2), 625–629 (1984)

    Article  MATH  Google Scholar 

  40. Weispfenning, V.: Quantifier eliminations and decision procedures for valued fields. In: Muller, G.H, Richter, M.M. (eds.) Models and Sets, Proceedings, Logic Colloquium Aachen, Part I, Lecture Notes in Mathematics, vol. 1103, pp. 414–472. Springer, Berlin (1984)

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Acknowledgements

We would like to thank the referee for his detailed reading, his useful (and numerous) remarks. We also would like to thank Zoé Chatzidakis for having drawn our attention to the Kolchin polynomials and Pablo Cubidès for discussing the cell decomposition theorem.

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  1. Département de Mathématique, Université de Mons, Le Pentagone, 20, Place du Parc, 7000, Mons, Belgium

    Françoise Point

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  1. Françoise Point
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Correspondence to Françoise Point.

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Research Director at the “Fonds de la Recherche Scientifique (F.R.S.-F.N.R.S.)”.

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Point, F. Definability of types and VC density in differential topological fields. Arch. Math. Logic 57, 809–828 (2018). https://doi.org/10.1007/s00153-017-0607-y

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