A NOTE ON b-MINIMAL LOCAL FIELDS

We prove criteria for axiom (b1) of [Cluckers, Loeser, b-minimality, http://www.dma.ens.fr/∼cluckers/]. We establish criteria for b-minimality when the auxiliary sorts are small. These criteria show that in many naturally occurring situations, for example in the situation of local fields, axiom (b1) is the only important axiom of b-minimality, in the sense that the other axioms of b-minimality follow from axiom (b1). Introduction. In [1], the notion of b-minimality is introduced, invoking three axioms, named (b1), (b2), and (b3). In a way, (b1) is an axiom stating that there is a nice cell decomposition, see Lemma’s 1 and 2 below and the definition of cells in [1]. Axioms (b2) and (b3) are included to guarantee a dimension theory with good properties. The major step made in the conception of b-minimality in [1] was to impose cell decomposition (with parameters) bluntly as an axiom of b-minimality, instead of searching for ingenious axioms implying cell decomposition. A similar move has been made in the theory of quantum groups: for a long time one was looking for axioms guaranteeing the existence of a unique Haar measure on such groups, until it was discovered that having a Haar measure was a very fruitful axiom in itself [5]. This is unlike other notions as o-minimality [3] or v-minimality [4] where cell decomposition follows as a consequence of the definitions and is not really imposed as an axiom. In this note, we show that in many situations the essential axiom of b-minimality is exactly the axiom (b1) which imposes cell decomposition. More precisely this is the case when the auxiliary sorts are small in comparison with the main sort, as is for example the case for the residue field and value group of a nonarchimedean local field compared to the local field itself, or for the real field relative to a countable auxiliary set. The cell decomposition of b-minimality is one that is useful for integration on nonarchimedean local fields, and goes back to Denef’s work on p-adic integrals [2]. We start by giving some criteria for axiom (b1). Terminology. We use the theory and terminology of [1]. By a (b1)-theory we mean a theory such that each model satisfies axiom (b1) of [1]. Recall that LB is the language with one predicate B and that TB is a theory saying that B is a nonempty subset of the Cartesian product of the main sort with some Cartesian product of the universes of some of the sorts. The fibers of the predicate B in the main sort form by definition the collection of balls in any given model.