Cell decomposition and classification of definable sets in poptimal fields
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.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 arXiv:
 arXiv:1412.2571
 Bibcode:
 2014arXiv1412.2571D
 Keywords:

 Mathematics  Logic