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Israel Journal of Mathematics

, Volume 204, Issue 1, pp 299–327 | Cite as

Valued difference fields and NTP2

  • Artem ChernikovEmail author
  • Martin Hils
Article

Abstract

We show that the theory of the non-standard Frobenius automorphism, acting on an algebraically closed valued field of equal characteristic 0, is NTP2. More generally, in the contractive as well as in the isometric case, we prove that a σ-Henselian valued difference field of equicharacteristic 0 is NTP2, provided both the residue difference field and the value group (as an ordered difference group) are NTP2.

Keywords

Abelian Group Difference Field Order Vector Space Theory Burden Residue Characteristic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Adl07]
    H. Adler, Strong theories, burden, and weight, preprint, 2007.Google Scholar
  2. [Adl08]
    H. Adler, An introduction to theories without the independence property, Archive for Mathematical Logic, to appear, 2008.Google Scholar
  3. [AvdD11]
    S. Azgın and L. van den Dries, Elementary theory of valued fields with a valuationpreserving automorphism, Journal of the Institute of Mathematics of Jussieu 10 (2011), 1–35.CrossRefzbMATHGoogle Scholar
  4. [Azg07]
    S. Azgın, Model theory of valued difference fields, PhD thesis, University of Illinois at Urbana-Champaign, 2007.Google Scholar
  5. [Azg10]
    S. Azgın, Valued fields with contractive automorphism and Kaplansky fields, Journal of Algebra 324 (2010), 2757–2785.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BMS07]
    L. Bélair, A. Macintyre and T. Scanlon, Model theory of the Frobenius on the Witt vectors, American Journal of Mathematics 129 (2007), 665–721.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BYC12]
    I. Ben Yaacov and A. Chernikov, An independence theorem for NTP 2 theories, submitted, arXiv:1207.0289v1, 2012.Google Scholar
  8. [CH99]
    Z. Chatzidakis and E. Hrushovski, Model theory of difference fields, Transactions of the American Mathematical Society 351 (1999), 2997–3071.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Che]
    A. Chernikov, Theories without the tree property of the second kind, Annals of Pure and Applied Logic 165 (2014), 695–723.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [CK12]
    A. Chernikov and I. Kaplan, Forking and dividing in NTP 2 theories, Journal of Symbolic Logic 77 (2012), 1–20.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Del81]
    F. Delon, Types sur C((X)), in Study Group on Stable Theories (Bruno Poizat), Second year: 1978/79 (French), Secrétariat Math., Paris, 1981, Ecp. No. 5, 29.Google Scholar
  12. [Gia11]
    G. Giabicani, Théorie de l’intersection en géométrie aux différences, PhD thesis, École Polytechnique, Palaiseau, 2011.Google Scholar
  13. [GS84]
    Y. Gurevich and P. H. Schmitt, The theory of ordered abelian groups does not have the independence property, Transactions of the American Mathematical Society 284 (1984), 171–182.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [HHM08]
    D. Haskell, E. Hrushovski and D. Macpherson, Stable Domination and Independence in Algebraically Closed Valued Fields, Lecture Notes in Logic, Vol. 30, Association for Symbolic Logic, Chicago, IL, 2008.zbMATHGoogle Scholar
  15. [Hod93]
    W. Hodges, Model Theory, Cambridge University Press, 1993.Google Scholar
  16. [HK06]
    E. Hrushovski and D. Kazhdan, Integration in valued fields, in Algebraic Geometry and Number Theory, Progress in Mathematics, Vol. 253, Birkhäuser, Boston, MA, 2006, pp. 261–405.CrossRefGoogle Scholar
  17. [HL11]
    E. Hrushovski and F. Loeser, Non-Archimedean tame topology and stably dominated types, Annals of Mathematics Studies, to appear.Google Scholar
  18. [Hru01]
    E. Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Annals of Pure and Applied Logic 112 (2001), 43–115.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [Hru02]
    E. Hrushovski, Valued fields with automorphisms, manuscript, 2002.Google Scholar
  20. [Hru04]
    E. Hrushovski, The elementary theory of the Frobenius automorphisms, arXiv:0406514v1, 2004.Google Scholar
  21. [Pal12]
    K. Pal, Multiplicative valued difference fields, Journal of Symbolic Logic 77 (2012), 545–579.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [Pas89]
    J. Pas, Uniform p-adic cell decomposition and local zeta functions, Journal für die Reine und Angewandte Mathematik 399 (1989), 137–172.MathSciNetzbMATHGoogle Scholar
  23. [Pre88]
    M. Prest, Model Theory and Modules, London Mathematical Society Lecture Note Series, Vol. 130, Cambridge University Press, Cambridge, 1988.CrossRefzbMATHGoogle Scholar
  24. [RZ60]
    A. Robinson and E. Zakon, Elementary properties of ordered abelian groups, Transactions of the American Mathematical Society 96 (1960), 222–236.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Sca00]
    T. Scanlon, A model complete theory of valued D-fields, Journal of Symbolic Logic 65 (2000), 1758–1784.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [Sca03]
    T. Scanlon, Quantifier elimination for the relative Frobenius, in Valuation Theory and its Applications, Vol. II (Saskatoon, SK, 1999), Fields Institute Communications, Vol. 33, American Mathematical Society, Providence, RI, 2003, pp. 323–352Google Scholar
  27. [She80]
    S. Shelah, Simple unstable theories, Annals of Mathematical Logic 19 (1980), 177–203.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [She90]
    S. Shelah, Classification Theory and the Number of Nonisomorphic Models, second edition, Studies in Logic and the Foundations of Mathematics, Vol. 92, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
  29. [She09]
    S. Shelah, Strongly dependent theories, arXiv:math/0504197v4, 2009.Google Scholar
  30. [Sim12]
    P. Simon, Lecture notes on NIP theories, Lecture Notes in Logics, to appear. arXiv: 1208.3944, 2012.Google Scholar
  31. [Wag00]
    F. O. Wagner, Simple Theories, Mathematics and its Applications, Vol. 503, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2014

Authors and Affiliations

  1. 1.Einstein Institute of Mathematics Edmond J. Safra Campus, Givat RamThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Institut de Mathématiques de Jussieu (UMR 7586 du CNRS)Université Paris Diderot Paris 7Paris Cedex 13France
  3. 3.École Normale Supérieure, Département de mathématiques et applicationsUMR 8553 du CNRSParis Cedex 05France

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