11 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
Strongly NIP almost real closed fields
The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP
field is either real closed, algebraically closed, or admits a non-trivial
definable henselian valuation, in the language of rings. We specialise this
conjecture to ordered fields in the language of ordered rings, which leads
towards a systematic study of the class of strongly NIP almost real closed
fields. As a result, we obtain a complete characterisation of this class.Comment: To appear in MLQ Math. Log. Q. A previous version of this preprint
was part of arXiv:1810.10377. arXiv admin note: text overlap with
arXiv:2010.1183
Definable valuations on ordered fields
We study the definability of convex valuations on ordered fields, with a
particular focus on the distinguished subclass of henselian valuations. In the
setting of ordered fields, one can consider definability both in the language
of rings and in the richer language of ordered rings
. We analyse and compare definability in both
languages and show the following contrary results: while there are convex
valuations that are definable in the language but
not in the language , any
-definable henselian valuation is already
-definable. To prove the latter, we show that the
value group and the ordered residue field of an ordered henselian valued field
are stably embedded (as an ordered abelian group, respectively as an ordered
field). Moreover, we show that in almost real closed fields any
-definable valuation is henselian.Comment: 17 page
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Model Theory: groups, geometry, and combinatorics
This conference was about recent interactions of model theory with combinatorics, geometric group theory and the theory of valued fields, and the underlying pure model-theoretic developments. Its aim was to report on recent results in the area, and to foster communication between the different communities