Skip Navigation


Journal of the London Mathematical Society Advance Access originally published online on May 13, 2008
Journal of the London Mathematical Society 2008 78(1):198-212; doi:10.1112/jlms/jdn024
This Article
Right arrow FREE Full Text (PDF) Freely available
Right arrow All Versions of this Article:
78/1/198    most recent
jdn024v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Cluckers, R.
Right arrow Articles by Robinson, Z.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2008 London Mathematical Society

Real closed fields with non-standard and standard analytic structure

R. Cluckers

Katholieke Universiteit Leuven
Departement Wiskunde
Celestijnenlaan 200B
B-3001 Leuven
Belgium
Current address:
École Normale Supérieure
Département de mathématiques et applications
45 rue d’Ulm
75230 Paris cedex 05
France
cluckers@ens.fr
www.wis.kuleuven.ac.be/algebra/Raf/

L. Lipshitz

Department of Mathematics
Purdue University
150 North University Street
West Lafayette, IN 47907-2067
USA
www.math.purdue.edu/~lipshitz/

Z. Robinson

Department of Mathematics
East Carolina University
Greenville, NC 27858-4353
USA
robinsonz@mail.ecu.edu

We consider the ordered field which is the completion of the Puiseux series field over R equipped with a ring of analytic functions on [–1, 1]n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures of Denef and van den Dries [Ann. of Math. 128 (1988) 79–138] and Lipshitz and Robinson [Bull. London Math. Soc. 38 (2006) 897–906]. We prove quantifier elimination and o-minimality in the corresponding language. We extend these constructions and results to rank n ordered fields Rn (the maximal completions of iterated Puiseux series fields). We generalize the example of Hrushovski and Peterzil [J. Symbolic Logic 72 (2007) 119–122] of a sentence which is not true in any o-minimal expansion of R (shown in [Bull. London Math. Soc. 38 (2006) 897–906] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences {sigma}n, true in Rn, but not true in any o-minimal expansion of any of the fields R, R1, ..., Rn–1.

2000 Mathematics Subject Classification 03C64, 32P05, 32B05, 32B20 (primary); 03C10, 03C98, 03C60, 14P15 (secondary).

The first author has been supported as a postdoctoral fellow by the Fund for Scientific Research — Flanders (Belgium) (F.W.O.) — and by the European Commission — Marie Curie European Individual Fellowship with contract number HPMF CT 2005-007121 — during the preparation of this paper; the second and third authors have been supported in part by NSF grant DMS-0401175.

Received October 23, 2006; revised February 26, 2008; published online May 13, 2008.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer:
Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.