Cell decomposition and dimension theory on p-optimal fields

Luck Darnière (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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, thus providing a new proof of the main result of van den Dries, Haskell and Macpherson [J. London Math. Soc, 59 (1999)]. The topological dimension introduced by Haskell and Macpherson in [JSL, 62, 4 (1997)] for strongly p--minimal fields is shown to be a "dimension function" in the sense of van den Dries [APAL 45 (1989)]. In particular it coincides in p-optimal fields with the topological rank. Finally we obtain for the restricted class of p2-optimal fields a preparation theorem for definable functions, from which it follows that infinite sets definable over such fields are isomorphic iff they have the same dimension.

Mots clés: p-optimal field; p-adic field; dimension of definable sets; cell decomposition; p-minimal field

Domaines: Logique [math.LO]

Luck Darnière. Cell decomposition and dimension theory on p-optimal fields. 2014. ⟨hal-01083119v2⟩