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
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: | https://doi.org/10.1017/jsl.2015.79 |