Françoise Delon

Separably closed fields

Separably closed fields are stable. When they are not algebraically closed, they are rather complicated from a model theoretic point of view: they are not super-stable, they admit no non trivial continuous rank and they have the dimensional order property. But they have a fairly good theory of types and independence, and interesting minimal types. Hrushovski used separably closed fields in his proof of the Mordell-Lang Conjecture for function fields in positive characteristic in the same way he used differentially closed fields in characteristic zero ([Hr 96], see [Bous] in this volume). In particular he proved that a certain class of minimal types, which he called thin, are Zariski geometries in the sense of [Mar] section 5. He then applied to these types the strong trichotomy theorem valid in Zariski geometries.

Some Model Theory for Almost Real Closed Fields

We study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k. ?

The theory of modules of separably closed fields 2

Abstract In Dellunde et al. (J. Symbolic Logic 67(3) (2002) 997–1015), we determined the complete theory T e of modules of separably closed fields of characteristic p and imperfection degree e , e ∈ ω ∪{∞}. Here, for 0≠ e ∈ ω , we describe the closed set of the Ziegler spectrum corresponding to T e . Further, we establish a correspondence between certain submodules and n -types and we investigate several notions of dimensions and their relationships with the Lascar rank. Finally, we show that T e has uniform p.p. elimination of imaginaries and deduce uniform weak elimination of imaginaries.

The undecidability of the theory of immediate pairs of Henselian valued fields

The theory of immediate pairs of Henselian valued fields, with a given residual theory (of characteristic zero) and a given theory of valuation group (nonzero), is undecidable and has completions.

Un Principe D'Ax-Kochen-Ershov Pour des Structures Intermediares Entre Groupes et Corps Values

An Ax-Kochen-Ershov principle for intermediate structures between valued groups and valued fields. We will consider structures that we call valued B -groups and which are of the form 〈 G, B , *, υ〉 where – G is an abelian group, – B is an ordered group, – υ is a valuation denned on G taking its values in B , – * is an action of B on G satisfying: ∀ x ϵ G ∀ b ∈ B υ( x * b ) = ν( x ) · b . The analysis of Kaplanski for valued fields can be adapted to our context and allows us to formulate an Ax-Kochen-Ershov principle for valued B -groups: we axiomatise those which are in some sense existentially closed and also obtain many of their model-theoretical properties. Let us mention some applications: 1. Assume that υ( x ) = υ( nx ) for every integer n ≠ 0 and x ϵ G, B is solvable and acts on G in such a way that, for the induced action, Z[ B ] ∖ {0} embeds in the automorphism group of G . Then 〈 G, B , *, υ〉 is decidable if and only if B is decidable as an ordered group. 2. Given a field k and an ordered group B , we consider the generalised power series field k (( B )) endowed with its canonical valuation. We consider also the following structure: where k (( B )) + is the additive group of k (( B )), S is a unary predicate interpreting { T b ∣ b ϵ B }, and ×↾ k (( B ))× S is the multiplication restricted to k (( B )) × S , structure which is a reduct of the valued field k (( B )) with its canonical cross section. Then our result implies that if B is solvable and decidable as an ordered group, then M is decidable. 3. A valued B –group has a residual group and our Ax-Kochen-Ershov principle remains valid in the context of expansions of residual group and value group. In particular, by adding a residual order we obtain new examples of solvable ordered groups having a decidable theory.

Sums of powers in function fields

Abstract When is a rational function on a smooth variety V whose values are sums of 2nth powers, itself a sum of 2nth powers in the field of rational functions? We investigate this property and some weaker form of it. The aim is to gain a better understanding of sums of powers in formally real function fields.

Formal power series

In this paper we survey some generalizations of formal Laurent power series to several indeterminates and we expound some of the fundamental logical results concerning fields of generalized power series. In connection with the above, we also present the notions of saturated model and of ultraproduct.

Corps Portant Un Nombre Fini de Valuations

Description du modele-compagnon des theories associees a un arbre fini quelconque de valuations