Darniere, L and Halupczok, I (2017) Cell decomposition and classification of definable sets in p-optimal fields. Journal of Symbolic Logic, 82 (1). pp. 120-136. ISSN 0022-4812
Abstract
We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to a p-adic one. For such fields K, we prove that every definable subset of K × K d whose fibers over K are inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions on p-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017, Association for Symbolic Logic. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | p-minimality; cell decomposition; definable sets; p-optimality |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 02 Feb 2016 13:54 |
Last Modified: | 29 Mar 2017 15:19 |
Published Version: | https://doi.org/10.1017/jsl.2015.79 |
Status: | Published |
Publisher: | Association for Symbolic Logic |
Identification Number: | 10.1017/jsl.2015.79 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:94338 |