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Preprint Number 796
796. Luck Darnière and immanuel Halupczok Cell decomposition and classification of definable sets in poptimal fields Email: Submission date: 13 November 2014. Revised version: 15 July 2015. Abstract: We prove that for poptimal fields (a very large subclass of pminimal 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 pminimality. Then we turn to strongly poptimal fields satisfying the Extreme Value Property (a property which in particular holds in fields which are elementarily equivalent to a padic 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 semialgebraic. Combining the two we get a preparation theorem for definable functions on poptimal 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. Mathematics Subject Classification: 03C07, 12J12 Keywords and phrases: Cell decomposition, pminimal, poptimal, dimension of definable sets Full text: arXiv 1412.2571: pdf, ps.
Version of 13 November 2014: pdf, ps,
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