67 research outputs found
Definable henselian valuations
In this note we investigate the question whether a henselian valued field
carries a non-trivial 0-definable henselian valuation (in the language of
rings). It follows from the work of Prestel and Ziegler that there are
henselian valued fields which do not admit a 0-definable non-trivial henselian
valuation. We give conditions on the residue field which ensure the existence
of a parameter-free definiton. In particular, we show that a henselian valued
field admits a non-trivial 0-definable valuation when the residue field is
separably closed or sufficiently non-henselian, or when the absolute Galois
group of the (residue) field is non-universal.Comment: 14 pages, revised versio
Model theory of finite-by-Presburger Abelian groups and finite extensions of -adic fields
We define a class of pre-ordered abelian groups that we call
finite-by-Presburger groups, and prove that their theory is model-complete. We
show that certain quotients of the multiplicative group of a local field of
characteristic zero are finite-by-Presburger and interpret the higher residue
rings of the local field. We apply these results to give a new proof of the
model completeness in the ring language of a local field of characteristic zero
(a result that follows also from work of Prestel-Roquette)
Henselianity in the language of rings
We consider four properties of a field K related to the existence of (de-finable) henselian valuations on K and on elementarily equivalent fields and study the implications between them. Surprisingly, the full pictures look very different in equichar-
acteristic and mixed characteristic
Monotone -convex -differential fields
Let be a complete, model complete o-minimal theory extending the theory
of real closed ordered fields and assume that is power bounded. Let be
a model of equipped with a -convex valuation ring and a
-derivation such that is monotone, i.e., weakly
contractive with respect to the valuation induced by . We show
that the theory of monotone -convex -differential fields, i.e., the
common theory of such , has a model completion, which is complete and
distal. Among the axioms of this model completion, we isolate an analogue of
henselianity that we call -henselianity. We establish an
Ax--Kochen/Ershov theorem and further results for monotone -convex
-differential fields that are -henselian.Comment: 26 page
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