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) cell decomposition and cell preparation for definable functions follow from methods going back to Denef's paper [Invent. Math, 77 (1984)]. We derive from it that the topological dimension introduced by Haskell and Macpherson in [JSL, 62, 4 (1997)] for P -minimal fields has even better properties in p-optimal fields, relatively to fundamental operations such as taking the boundary of a definable set, or the fibers of a definable function. As a consequence the topological dimension coincides with the topological rank, and every two infinite definable sets are isomorphic if and only if they have the same dimension.

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

Domaines: Logique [math.LO]

Luck Darnière. Cell decomposition and dimension theory on p-optimal fields. 2014. ⟨hal-01083119v1⟩