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

2014/12/08 by Luck Darnière, Darnière, Luck, Immanuel Halupczok +1
Mathematics · #FOS: Mathematics #Logic (math.LO) #math.LO

paper · pdf · doi:10.48550/arxiv.1412.2571

arxiv created 2016/02/05 · arxiv updated 2016/02/08

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.

Related

• Definable completeness of P-minimal fields and applications
• Cell Decomposition for semibounded p-adic sets
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• Presburger sets and p-minimal fields
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