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

Luck Darnière; Immanuel Halupczok

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arXiv · 1412.2571

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

Luck Darnière·Immanuel Halupczok

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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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BibTeX RIS

Luck Darnière, Immanuel Halupczok. 2014-12-08. Cell decomposition and classification of definable sets in p-optimal fields. https://arxiv.org/abs/1412.2571

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