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Cell decomposition and classification of definable sets in p-optimal fields

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 [Invent. Math, 77 (1984)]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-optimal 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 KxK^d whose fibers are inverse images by the valuation of subsets of the value group, are semi-algebraic. 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 isomorphic iff they have the same dimension.
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Submitted on : Friday, February 5, 2016 - 1:38:19 PM
Last modification on : Friday, January 7, 2022 - 3:56:40 AM
Long-term archiving on: : Saturday, November 12, 2016 - 11:02:01 AM

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  • HAL Id : hal-01083119, version 4
  • ARXIV : 1412.2571

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Luck Darnière, Immanuel Halupczok. Cell decomposition and classification of definable sets in p-optimal fields. The Journal of Symbolic Logic, Association for Symbolic Logic, 2017, 82 (1), pp.120-136. ⟨hal-01083119v4⟩

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