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STRONGLY MINIMAL REDUCTS OF VALUED FIELDS

Published online by Cambridge University Press:  29 June 2016

PIOTR KOWALSKI  and
SERGE RANDRIAMBOLOLONA
PIOTR KOWALSKI
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI WROCŁAW, POLANDE-mail: [email protected]: http://www.math.uni.wroc.pl/∼pkowa/
SERGE RANDRIAMBOLOLONA
Affiliation:
GALATASARAY ÜNIVERSITESI ISTANBUL, TURKEYE-mail: [email protected]
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Abstract

We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

Keywords

03C4512J25Strongly minimal structuresvalued fields
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 510 - 523
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Abhyankar, S. S., Local analytic geometry . Pure and Applied Mathematics, Elsevier Science, London, 1964.Google Scholar
Delon, Françoise, Corps C-minimaux, en l’honneur de François Lucas. Annales de la faculté des Sciences de Toulouse , Mathématiques, Série 6, XXI, pp. 413–434, 2012.CrossRefGoogle Scholar
Haskell, D., Hrushovski, E., and Macpherson, D., Definable sets in algebraically closed valued fields: elimination of imaginaries . Journal für die reine und angewandte Mathematik, vol. 597 (2006), pp. 175–236.Google Scholar
Haskell, D., Hrushovski, E., and Macpherson, D., Stable Domination and Independence in Algebraically Closed Valued Fields , Lecture Notes in Logic, Cambridge University Press, 2008.Google Scholar
Haskell, D., and Macpherson, D., Cell decompositions of C-minimal structures . Annals of Pure and Applied Logic, vol. 66 (1994), no. 2, pp. 113–162.CrossRefGoogle Scholar
Hasson, Assaf, and Kowalski, Piotr, Strongly minimal expansions of (ℂ, +) definable in o-minimal fields . Proceedings of the London Mathematical Society, vol. 97 (2008), no. 1, pp. 117–154.Google Scholar
Hasson, Assaf, Onshuus, Alf, and Peterzil, Ya’acov, Definable one dimensional structures in o-minimal theories . Israel Journal of Mathematics, vol. 179 (2010), pp. 297–361.CrossRefGoogle Scholar
Holly, Jan E., Canonical forms for definable subsets of algebraically closed and real closed valued fields, this Journal, vol. 60 (1995), no. 3, pp. 843–860.Google Scholar
Hrushovski, Ehud, Strongly minimal expansions of algebraically closed fields . Israel Journal of Mathematics, vol. 79 (1992), pp. 129–151.Google Scholar
Hrushovski, Ehud, Valued fields, metastable groups, preprint, available on http://www.ma.huji.ac.il/ehud/mst2013nov4.pdf.Google Scholar
Hrushovski, Ehud and Zilber, Boris, Zariski geometries . Journal of the American Mathematical Society, vol. 9 (1996), no. 1, pp. 1–55.Google Scholar
Cubides Kovacsics, P., An introduction to C-minimal structures and their cell , Valuation Theory and Interaction (Campillo, Antonio, Kuhlmann, Franz-Viktor, and Teissier, Bernard, editor), Series of Congress Reports of the EMS, pp. 167–207, 2014.Google Scholar
Lipshitz, Leonard, and Robinson, Zachary, Uniform properties of rigid subanalytic sets . Transactions of the American Mathematical Society, vol. 357 (2005), pp. 4349–4377.Google Scholar
Maalouf, Fares, Additive reducts of algebraically closed valued fields . Forum Mathematicum, vol. 27 (2015), pp. 877–895.CrossRefGoogle Scholar
Marker, David, and Pillay, Anand, Reducts of (ℂ, +,⋅) which contain +, this Journal, vol. 55 (1990), no. 3, pp. 1243–1251.Google Scholar
Otero, Margarita, Peterzil, Ya’acov, and Pillay, Anand, On groups and rings definable in o-minimal expansions of real closed fields . Bulletin of the London Mathematical Society, vol. 28 (1996), no. 1, pp. 7–14.CrossRefGoogle Scholar
Pas, Johan, Uniform p-adic cell decomposition and local zeta functions . Journal für die reine und Angewandte Mathematik, vol. 399 (1989), pp. 137–172.Google Scholar
Peterzil, Ya’acov, and Starchenko, Sergei, Expansions of algebraically closed fields in o-minimal structures . Selecta Mathematica (N.S.), vol. 7 (2001), no. 3, pp. 409–445.CrossRefGoogle Scholar
Pillay, Anand, Geometric stability theory , Oxford Logic Guides, vol. 32, The Clarendon Press Oxford University Press, New York, 1996. Oxford Science Publications.Google Scholar
Poizat, Bruno, Stable groups , Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, Providence, RI, 2001. Translated from the 1987 French original by Moses Gabriel Klein.Google Scholar
Ribenboim, P., The Theory of Classical Valuations, Monographs in Mathematics, Springer, New York, 1999.Google Scholar
Robinson, A., Complete Theories, Studies in logic and the foundations of mathematics, North-Holland, Amsterdam, 1956.Google Scholar
Wagner, Frank, A note on one-basedness, this Journal, vol. 69 (2004), no. 2, pp. 34–38.Google Scholar
Zilber, Boris, The structure of models of uncountably categorical theories , Proceedings of the International Congress of Mathematicians, vol. 1, pp. 359–368, PWN - North-Holland, Amsterdam, 1984.Google Scholar