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Real closed valued fields with analytic structure

Part of: Non-Archimedean analysis Model theory

Published online by Cambridge University Press:  05 December 2019

Pablo Cubides Kovacsics  and
Deirdre Haskell
Pablo Cubides Kovacsics
Affiliation:
TU Dresden, Fachrichtung Mathematik, Institut für Algebra, 01062Dresden, Germany ([email protected])
Deirdre Haskell
Affiliation:
Department of Mathematics and Statistics, McMaster University, 1280 Main St W, Hamilton, ON L8S 4K1, Canada ([email protected])
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Abstract

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We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.

Keywords

real closed valued fieldsseparated analytic structureoverconvergent power serieso-minimalityweak o-minimalityC-minimality

MSC classification

Primary: 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem) 32B05: Analytic algebras and generalizations, preparation theorems 03C10: Quantifier elimination, model completeness and related topics 03C64: Model theory of ordered structures; o-minimality
Secondary: 14P10: Semialgebraic sets and related spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 249 - 261
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Edinburgh Mathematical Society 2019

References

1.Cherlin, G. and Dickmann, M. A., Real closed rings II. Model theory, Ann. Pure Appl. Logic 25(3) (1983), 213–231.CrossRefGoogle Scholar
2.Cluckers, R., Analytic p-adic cell decomposition and integrals, Trans. Am. Math. Soc. 356(4) (2004), 1489–1499.10.1090/S0002-9947-03-03458-5CrossRefGoogle Scholar
3.Cluckers, R. and Lipshitz, L., Fields with analytic structure, J. Eur. Math. Soc. (JEMS) 13 (2011), 1147–1223.CrossRefGoogle Scholar
4.Cluckers, R., Lipshitz, L. and Robinson, Z., Analytic cell decomposition and analytic motivic integration, Ann. Sci. École Norm. Sup. (4) 39(4) (2006), 535–568.CrossRefGoogle Scholar
5.Cluckers, R., Lipshitz, L. and Robinson, Z., Real closed fields with non-standard and standard analytic structure, J. Lond. Math. Soc. (2) 78(1) (2008), 198–212.10.1112/jlms/jdn024CrossRefGoogle Scholar
6.Denef, J. and van den Dries, L., p-adic and real subanalytic sets, Ann. of Math. (2) 128(1) (1988), 79–138.CrossRefGoogle Scholar
7.Lipshitz, L., Rigid subanalytic sets, Am. J. Math. 115(1) (Feb. 1993), 77–108.CrossRefGoogle Scholar
8.Lipshitz, L. and Robinson, Z., One-dimensional fibers of rigid subanalytic sets, J. Symbolic Logic 63 (Mar. 1998), 83–88.CrossRefGoogle Scholar
9.Lipshitz, L. and Robinson, Z., Rings of separated power series and quasi-affinoid geometry, Astérisque 264(264) (2000), vi+171.Google Scholar
10.Lipshitz, L. and Robinson, Z., Overconvergent real closed quantifier elimination, Bull. London Math. Soc. 38(6) (2006), 897–906.10.1112/S0024609306018832CrossRefGoogle Scholar
11.Schoutens, H., Rigid subanalytic sets, Compos. Math. 94(3) (1994), 269–295.Google Scholar
12.van den Dries, L., Haskell, D. and Macpherson, D., One-dimensional p-adic subanalytic sets, J. London Math. Soc. 59(1) (1999), 1–20.10.1112/S0024610798006917CrossRefGoogle Scholar
13.van den Dries, L. and Lewenberg, A. H., T-convexity and tame extensions, J. Symbolic Logic 60(1) (1995), 74–102.CrossRefGoogle Scholar
14.van den Dries, L., Macintyre, A. and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140(1) (1994), 183–205.10.2307/2118545CrossRefGoogle Scholar
15.van den Dries, L., Macintyre, A. and Marker, D., Logarithmic-exponential power series, J. London Math. Soc. (2) 56(3) (1997), 417–434.10.1112/S0024610797005437CrossRefGoogle Scholar
16.van den Dries, L., Macintyre, A., and Marker, D., Logarithmic-exponential series, Ann. Pure Appl. Logic 111(1–2) (2001), 61–113.10.1016/S0168-0072(01)00035-5CrossRefGoogle Scholar