Groups and fields with $\operatorname {NTP}_{2}$
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- by Artem Chernikov, Itay Kaplan and Pierre Simon
- Proc. Amer. Math. Soc. 143 (2015), 395-406
- DOI: https://doi.org/10.1090/S0002-9939-2014-12229-5
- Published electronically: August 19, 2014
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Abstract:
$\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in $\operatorname {NTP}_{2}$ structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any $\operatorname {NTP}_{2}$ field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky.References
- Hans Adler, Strong theories, burden, and weight, preprint (2007).
- Ricardo de Aldama, A result on definable groups without the independence property, Bulletin of Symbolic Logic, to appear.
- James Ax and Simon Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630. MR 184930, DOI 10.2307/2373065
- James Ax, On the undecidability of power series fields, Proc. Amer. Math. Soc. 16 (1965), 846. MR 177890, DOI 10.1090/S0002-9939-1965-0177890-2
- Salih Azgin, Valued fields with contractive automorphism and Kaplansky fields, J. Algebra 324 (2010), no. 10, 2757–2785. MR 2725200, DOI 10.1016/j.jalgebra.2010.08.003
- Luc Bélair, Types dans les corps valués munis d’applications coefficients, Illinois J. Math. 43 (1999), no. 2, 410–425 (French, with English summary). MR 1703196
- Itaï Ben Yaacov and Artem Chernikov, An independence theorem for NTP2 theories, arXiv:1207.0289 (2012).
- Zoé Chatzidakis, Simplicity and independence for pseudo-algebraically closed fields, Models and computability (Leeds, 1997) London Math. Soc. Lecture Note Ser., vol. 259, Cambridge Univ. Press, Cambridge, 1999, pp. 41–61. MR 1721163, DOI 10.1017/CBO9780511565670.004
- Zoé Chatzidakis, Independence in (unbounded) PAC fields, and imaginaries, http://www.logique.jussieu.fr/~zoe/papiers/Leeds08.pdf (2008).
- Artem Chernikov, Theories without the tree property of the second kind, Ann. Pure Appl. Logic 165 (2014), no. 2, 695–723. MR 3129735, DOI 10.1016/j.apal.2013.10.002
- Artem Chernikov and Martin Hils, Valued difference fields and ${NTP_2}$, Israel Journal of Mathematics, doi: 10.1007/s11856-014-1094-z, 2014, pages 1–29.
- Artem Chernikov and Itay Kaplan, Forking and dividing in $\textrm {NTP}_2$ theories, J. Symbolic Logic 77 (2012), no. 1, 1–20. MR 2951626, DOI 10.2178/jsl/1327068688
- Alfred Dolich, John Goodrick, and David Lippel, Dp-minimality: basic facts and examples, Notre Dame J. Form. Log. 52 (2011), no. 3, 267–288. MR 2822489, DOI 10.1215/00294527-1435456
- Ido Efrat, The elementary theory of free pseudo $p$-adically closed fields of finite corank, J. Symbolic Logic 56 (1991), no. 2, 484–496. MR 1133080, DOI 10.2307/2274695
- Michael D. Fried and Moshe Jarden, Field arithmetic, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2005. MR 2102046
- Dan Haran and Moshe Jarden, The absolute Galois group of a pseudo $p$-adically closed field, J. Reine Angew. Math. 383 (1988), 147–206. MR 921990, DOI 10.1515/crll.1988.383.147
- Ehud Hrushovski, The elementary theory of the Frobenius automorphisms, arXiv:math/0406514.
- Ehud Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25 (2012), no. 1, 189–243. MR 2833482, DOI 10.1090/S0894-0347-2011-00708-X
- Itay Kaplan, Thomas Scanlon, and Frank O. Wagner, Artin-Schreier extensions in NIP and simple fields, Israel J. Math. 185 (2011), 141–153. MR 2837131, DOI 10.1007/s11856-011-0104-7
- Itay Kaplan and Saharon Shelah, Chain conditions in dependent groups, Ann. Pure Appl. Logic 164 (2013), no. 12, 1322–1337. MR 3093393, DOI 10.1016/j.apal.2013.06.014
- Byunghan Kim, Simple first order theories, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–University of Notre Dame. MR 2694252
- Krzysztof Krupiński and Anand Pillay, On stable fields and weight, J. Inst. Math. Jussieu 10 (2011), no. 2, 349–358. MR 2787692, DOI 10.1017/S1474748010000228
- Urs-Martin Künzi, Corps multiplement pseudo-$p$-adiquement clos, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 4, 205–208 (French, with English summary). MR 1006730
- Urs-Martin Kyuntsi, Decidable theories of pseudo-$p$-adically closed fields, Algebra i Logika 28 (1989), no. 6, 643–669, 743 (Russian); English transl., Algebra and Logic 28 (1989), no. 6, 421–438 (1990). MR 1087579, DOI 10.1007/BF01980234
- Cédric Milliet, Definable envelopes in groups with simple theory, http://hal.archives-ouvertes.fr/hal-00657716/fr/ (English).
- Anand Pillay, Definability and definable groups in simple theories, J. Symbolic Logic 63 (1998), no. 3, 788–796. MR 1649061, DOI 10.2307/2586712
- Alexander Prestel, Pseudo real closed fields, Set theory and model theory (Bonn, 1979) Lecture Notes in Math., vol. 872, Springer, Berlin-New York, 1981, pp. 127–156. MR 645909
- Alexander Prestel, On the axiomatization of PRC-fields, Methods in mathematical logic (Caracas, 1983) Lecture Notes in Math., vol. 1130, Springer, Berlin, 1985, pp. 351–359. MR 799048, DOI 10.1007/BFb0075318
- Alexander Prestel, Pseudo real closed fields, Séminaire sur les Structures Algébriques Ordonnées, Vol. I, Publ. Math. Univ. Paris VII, vol. 32, Univ. Paris VII, Paris, 1990, pp. 33–35. MR 1071779
- S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
- Saharon Shelah, Simple unstable theories, Ann. Math. Logic 19 (1980), no. 3, 177–203. MR 595012, DOI 10.1016/0003-4843(80)90009-1
- Saharon Shelah, Dependent first order theories, continued, Israel J. Math. 173 (2009), 1–60. MR 2570659, DOI 10.1007/s11856-009-0082-1
Bibliographic Information
- Artem Chernikov
- Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel
- Address at time of publication: L’équipe de Logique Mathématique, IMJ-PRG, Université Paris Diderot-Paris 7, UFR de Mathématiques, case 7012, 75205 Paris Cedex 13, France
- Email: art.chernikov@gmail.com
- Itay Kaplan
- Affiliation: Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
- Address at time of publication: Institute of Mathematics, Hebrew University (The Edmond J. Safra Campus), Givat Ram, Jerusalem 91904, Israel
- MR Author ID: 886730
- Email: itay.kaplan@uni-muenster.de
- Pierre Simon
- Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel
- Address at time of publication: Université Claude Bernard-Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
- MR Author ID: 942320
- Email: pierre.simon@normalesup.org
- Received by editor(s): December 31, 2012
- Received by editor(s) in revised form: February 26, 2013
- Published electronically: August 19, 2014
- Additional Notes: The first author was partially supported by the [European Community’s] Seventh Framework Programme [FP7/2007-2013] under grant agreement No. 238381
The second author was supported by SFB 878 - Communicated by: Julia Knight
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 395-406
- MSC (2010): Primary 03C45, 03C60
- DOI: https://doi.org/10.1090/S0002-9939-2014-12229-5
- MathSciNet review: 3272764