Abstract
In this paper, we undertake a systematic model- and valuation-theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah–Hasson Conjecture (specialized to ordered fields) and provide an example limiting its valuation-theoretic conclusions.
Funding statement: The first author was supported by a doctoral scholarship of Studienstiftung des deutschen Volkes as well as of Carl-Zeiss-Stiftung and an Independent Research Grant of Zukunftskolleg, Universität Konstanz.
Acknowledgements
We started this research at the Model Theory, Combinatorics and Valued fields Trimester at the Institut Henri Poincaré in March 2018. All three authors wish to thank the IHP for its hospitality, and Immanuel Halupczok, Franziska Jahnke, Vincenzo Mantova and Florian Severin for discussions. We also thank Assaf Hasson for helpful comments on a previous version of this work and Vincent Bagayoko for giving an answer to a question leading to Proposition 4.7. Moreover, we thank the anonymous referee for providing several helpful comments and references.
References
[1] N. L. Alling, Foundations of Analysis Over Surreal Number Fields, North-Holland Math. Stud. 141, North-Holland, Amsterdam, 1987. Search in Google Scholar
[2] D. Biljakovic, M. Kochetov and S. Kuhlmann, Primes and irreducibles in truncation integer parts of real closed fields, Logic in Tehran, Lect. Notes Log. 26, AK Peters, Wellesley (2006), 42–64. 10.1201/9781439865873-3Search in Google Scholar
[3] P. F. Conrad, Regularly ordered groups, Proc. Amer. Math. Soc. 13 (1962), 726–731. 10.1090/S0002-9939-1962-0146272-9Search in Google Scholar
[4] F. Delon and R. Farré, Some model theory for almost real closed fields, J. Symb. Log. 61 (1996), no. 4, 1121–1152. 10.2307/2275808Search in Google Scholar
[5] K. Dupont, A. Hasson and S. Kuhlmann, Definable valuations induced by multiplicative subgroups and NIP fields, Arch. Math. Logic 58 (2019), no. 7–8, 819–839. 10.1007/s00153-019-00661-2Search in Google Scholar
[6] A. J. Engler and A. Prestel, Valued Fields, Springer Monogr. Math., Springer, Berlin, 2005. Search in Google Scholar
[7] P. Erdös, L. Gillman and M. Henriksen, An isomorphism theorem for real-closed fields, Ann. of Math. (2) 61 (1955), 542–554. 10.2307/1969812Search in Google Scholar
[8] R. Farré, A transfer theorem for Henselian valued and ordered fields, J. Symb. Log. 58 (1993), no. 3, 915–930. 10.2307/2275104Search in Google Scholar
[9] A. Fehm and F. Jahnke, Recent progress on definability of Henselian valuations, Ordered Algebraic Structures and Related Topics, Contemp. Math. 697, American Mathematical Society, Providence (2017), 135–143. 10.1090/conm/697/14049Search in Google Scholar
[10] Y. Halevi and A. Hasson, Strongly dependent ordered abelian groups and Henselian fields, Israel J. Math. 232 (2019), no. 2, 719–758. 10.1007/s11856-019-1885-3Search in Google Scholar
[11] K. Hauschild, Cauchyfolgen höheren Typus in angeordneten Körpern, Z. Math. Logik Grundlagen Math. 13 (1967), 55–66. 10.1002/malq.19670130303Search in Google Scholar
[12] J. Hong, Definable non-divisible Henselian valuations, Bull. Lond. Math. Soc. 46 (2014), no. 1, 14–18. 10.1112/blms/bdt074Search in Google Scholar
[13] F. Jahnke and J. Koenigsmann, Definable Henselian valuations, J. Symb. Log. 80 (2015), no. 1, 85–99. 10.1017/jsl.2014.64Search in Google Scholar
[14] F. Jahnke and J. Koenigsmann, Defining coarsenings of valuations, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 3, 665–687. 10.1017/S0013091516000341Search in Google Scholar
[15] F. Jahnke, P. Simon and E. Walsberg, Dp-minimal valued fields, J. Symb. Log. 82 (2017), no. 1, 151–165. 10.1017/jsl.2016.15Search in Google Scholar
[16] W. Johnson, The canonical topology on dp-minimal fields, J. Math. Log. 18 (2018), no. 2, Article ID 1850007. 10.1142/S0219061318500071Search in Google Scholar
[17] W. Johnson, Dp-finite fields VI: the dp-finite Shelah conjecture, preprint (2020), https://arxiv.org/abs/2005.13989v1. Search in Google Scholar
[18] M. Knebusch and M. J. Wright, Bewertungen mit reeller Henselisierung, J. Reine Angew. Math. 286(287) (1976), 314–321. 10.1515/crll.1976.286-287.314Search in Google Scholar
[19] L. S. Krapp, S. Kuhlmann and G. Lehéricy, On strongly NIP ordered fields and definable convex valuations, preprint (2019), https://arxiv.org/abs/1810.10377v4. Search in Google Scholar
[20] L. S. Krapp, S. Kuhlmann and G. Lehéricy, Strongly NIP almost real closed fields, preprint (2021), https://arxiv.org/abs/2010.14770v2; to appear in MLQ Math. Log. Q. 10.1002/malq.202000060Search in Google Scholar
[21] S. Kuhlmann, Ordered Exponential Fields, Fields Inst. Monogr. 12, American Mathematical Society, Providence, 2000. 10.1090/fim/012Search in Google Scholar
[22] K. McKenna, New facts about Hilbert’s seventeenth problem, Model Theory and Algebra: A Memorial Tribute to Abraham Robinson, Lecture Notes in Math. 498, Springer, Berlin (1975), 220–230. 10.1007/BFb0080982Search in Google Scholar
[23] M.-H. Mourgues and J. P. Ressayre, Every real closed field has an integer part, J. Symb.Log. 58 (1993), no. 2, 641–647. 10.2307/2275224Search in Google Scholar
[24] A. Prestel, Definable Henselian valuation rings, J. Symb. Log. 80 (2015), no. 4, 1260–1267. 10.1017/jsl.2014.52Search in Google Scholar
[25] A. Robinson and E. Zakon, Elementary properties of ordered abelian groups, Trans. Amer. Math. Soc. 96 (1960), 222–236. 10.1090/S0002-9947-1960-0114855-0Search in Google Scholar
[26] J. Robinson, Definability and decision problems in arithmetic, J. Symb. Log. 14 (1949), 98–114. 10.2307/2266510Search in Google Scholar
[27] S. Shelah, Strongly dependent theories, Israel J. Math. 204 (2014), no. 1, 1–83. 10.1007/s11856-014-1111-2Search in Google Scholar
[28] P. Simon, A Guide to NIP Theories, Lect. Notes Log. 44, Cambridge University, Cambridge, 2015. 10.1017/CBO9781107415133Search in Google Scholar
[29] T. M. Viswanathan, Ordered fields and sign-changing polynomials, J. Reine Angew. Math. 296 (1977), 1–9. 10.1515/crll.1977.296.1Search in Google Scholar
[30] E. Zakon, Generalized archimedean groups, Trans. Amer. Math. Soc. 99 (1961), 21–40. 10.1090/S0002-9947-1961-0120294-XSearch in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
