Cell Decomposition and Classification of Definable Sets in P-Optimal Fields

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 [Den84]. We derive from it the existence of definable Skolem functions and strong p-minimality, thus providing a new proof of the main result of [vdDHM99]. Then we turn to strongly p-optimal field 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 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.