## Dp-minimal valued fields.(English)Zbl 1385.03040

In this paper, the authors investigate the algebraic structure of dp-minimal valued fields. Dp-minimal theories can be thought of as a one-dimensional version of NIP theories; specifically, they are the theories in which the dp-rank of every type in one variable is at most one. Dp-minimality is a common generalization of C-minimality, o-minimality, and P-minimality, and hence the class of dp-minimal valued fields includes algebraically closed, real closed, and $$p$$-adically closed fields.
In [Fun with fields. Berkeley, CA: University of California (Diss.) (2016)], W. A. Johnson classified all dp-minimal fields up to elementary equivalence (see also [“On dp-minimal fields”, Preprint, arXiv:1507.02745]). Johnson constructs a valuation on any dp-minimal field that is not strongly minimal, and shows that the constructed valuation is always Henselian. This paper was developed independently of Johnson’s work [loc. cit.], and proves many of the same results using different methods. The authors provide a detailed comparison between their work and Johnson’s work [loc. cit.] in the introduction.
Section 3 of this paper generalizes results of J. Goodrick [J. Symb. Log. 75, No. 1, 221–238 (2010; Zbl 1184.03035)] and P. Simon [J. Symb. Log. 76, No. 2, 448–460 (2011; Zbl 1220.03037)] from the setting of dp-minimal ordered structures to the setting of dp-minimal structures with a definable uniform structure. A uniform structure is a collection of subsets which are able to replace intervals in the results of Goodrick [loc. cit.] and Simon [loc. cit.]; when considering valued fields, the uniform structure will be a collection of balls, that is, sets of the form $$\{x\in K : v(x-a) > \gamma\}$$.
The main results of Section 3 are Propositions 3.6 and 3.9: if $$M$$ is a structure with both a divisble abelian group operation and a uniform structure (for instance, if $$M$$ is a valued field), then every infinite definable subset of $$M$$ has non-empty interior and the image of any set with non-empty interior under a definable finite-to-one function also has non-empty interior.
Section 4 uses the results of Section 3 to prove the main result of the paper: every dp-minimal valued field is Henselian. Finally, Sections 5 and 6 classify dp-minimal ordered fields up to elementary equivalence: any such field is elementarily equivalent to a Hahn series field with residue field $$\mathbb R$$ and nonsingular value group (meaning $$\Gamma/p\Gamma$$ is finite for all primes $$p$$).
The paper is very accessible, and only assumes a passing familiarity with dp-minimality. The content on valuation theory makes more demands of the reader, but still clearly states and references any results that are not proved within the paper.

### MSC:

 03C60 Model-theoretic algebra 03C45 Classification theory, stability, and related concepts in model theory 03C64 Model theory of ordered structures; o-minimality 12J12 Formally $$p$$-adic fields 12J10 Valued fields 12J15 Ordered fields 12L12 Model theory of fields

### Citations:

Zbl 1184.03035; Zbl 1220.03037
Full Text:

### References:

 [1] Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., and Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property, Transaction of the American Mathematical Society, (2016), 5889-5949. · Zbl 1423.03119 [2] Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., and Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property, Notre Dame Journal of Formal Logic, vol. 54 (2013), no. 3, 4, pp. 311-363. doi:10.1215/00294527-2143862 · Zbl 1436.03185 [3] Chernikov, A. and Simon, P., Henselian valued fields and inp-minimality, preprint, 2015. [4] Cluckers, R. and Halupczok, I., Quantifier elimination in ordered abelian groups. Confluentes Mathematici, vol. 3 (2011), no. 4, pp. 587-615. doi:10.1142/S1793744211000473 · Zbl 1246.03059 [5] Engler, A. J. and Prestel, A., Valued Fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. · Zbl 1128.12009 [6] Goodrick, J., A monotonicity theorem for dp-minimal densely ordered groups, this Journal, vol. 75 (2010), no. 1, pp. 221-238. · Zbl 1184.03035 [7] Guingona, V., On vc-minimal fields and dp-smallness. Archive for Mathematical Logic, vol. 53 (2014), no. 5, 6, pp. 503-517. doi:10.1007/s00153-014-0376-9 · Zbl 1354.03047 [8] Jahnke, F. and Koenigsmann, J., Uniformly defining p-henselian valuations. Annals of Pure and Applied Logic, vol. 166 (2015), no. 7, 8, pp. 741-754. doi:10.1016/j.apal.2015.03.003 · Zbl 1372.03077 [9] Johnson, W., On dp-minimal fields, preprint, 2015. [10] Kaplan, I., Scanlon, T., and Wagner, F. O., Artin-Schreier extensions in NIP and simple fields. Israel Journal of Mathematics, vol. 185 (2011), pp. 141-153. doi:10.1007/s11856-011-0104-7 · Zbl 1261.03120 [11] Kudaĭbergenov, K. Zh., Weakly quasi-o-minimal models. Turkish Journal of Mathematics, vol. 13 (2010), no. 1, pp. 156-168. · Zbl 1249.03074 [12] Macintyre, A., Mckenna, K., and Van Den Dries, L., Elimination of quantifiers in algebraic structures. Advances in Mathematics, vol. 47 (1983), no. 1, pp. 74-87. doi:10.1016/0001-8708(83)90055-5 · Zbl 0531.03016 [13] Macpherson, D., Marker, D., and Steinhorn, C., Weakly o-minimal structures and real closed fields. Transactions of the American Mathematical Society, vol. 352 (2000), pp. 5435-5483. doi:10.1090/S0002-9947-00-02633-7 · Zbl 0982.03021 [14] Onshuus, A. and Usvyatsov, A., On dp-minimality, strong dependence and weight, this Journal, vol. 76 (2011), no. 3, pp. 737-758. · Zbl 1245.03053 [15] Prestel, A. and Delzell, C. N., Mathematical Logic and Model Theory, Universitext, Springer, 2011. · Zbl 1241.03001 [16] Prestel, A. and Ziegler, M., Model-theoretic methods in the theory of topological fields. Journal für die Reine und Angewandte Mathematik, vol. 299 (1978), no. 300, pp. 318-341. · Zbl 0367.12014 [17] Simon, P., On dp-minimal ordered structures, this Journal, vol. 76 (2011), pp. 448-460. · Zbl 1220.03037 [18] Simon, P., Dp-minimality: Invariant types and dp-rank, this Journal, vol. 79 (2014), pp. 1025-1045. · Zbl 1353.03036 [19] Simon, P., A Guide to NIP Theories, Lecture Notes in Logic. Cambridge University Press, Cambridge, 2015.
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