Auteurs : Krapp, Lothar Sebastian (Auteur de la Conférence)
CIRM (Editeur )
Résumé :
A valuation $v$ on a field $K$ is said to be definable (in a specified language) if its corresponding valuation ring is a definable subset of $K$. Historically, the study of definable valuations on certain fields was motivated by the general analysis of definable subsets of fields and related decidability questions, but has also re-emerged lately in the context of classifying NIP fields. In my talk, I will present some recent progress in the study of definable valuations on ordered fields ([1] to [4]), where definability is considered in the language of rings as well as the richer language of ordered rings. Within this framework, the focus lies on convex valuations, that is, valuations whose valuation ring is convex with respect to the linear ordering on the field. The most important examples of such valuations are the henselian ones, which are convex with respect to any linear ordering on the field. I will present topological conditions on the value group and the residue field ensuring the definability of the corresponding valuation. Moreover, I will outline some definability and non-definability results in the context of specific classes of ordered fields such as t-henselian, almost real closed, and strongly dependent ones.
Keywords : definable valuations; ordered fields; convex valuations; henselian valuations; almost real closed fields
Codes MSC :
03C64
- Model theory of ordered structures; o-minimality
12J10
- Valued fields
12J25
- Non-Archimedean valued fields, See also {30G06, 32P05, 46S10, 47S10}
13F25
- Formal power series rings, See also {13J05}
13J15
- Henselian rings, See also {13B40}
13J30
- Real algebra
Ressources complémentaires :
https://webusers.imj-prg.fr/~zoe.chatzidakis/CIRM/Krapp.pdf
|
Informations sur la Rencontre
Nom de la rencontre : Model theory of valued fields / Théorie des modèles des corps valués Organisateurs de la rencontre : Chatzidakis, Zoé ; Jahnke, Franziska ; Rideau-Kikuchi, Silvain Dates : 29/05/2023 - 02/06/2023
Année de la rencontre : 2023
URL Congrès : https://conferences.cirm-math.fr/2761.html
DOI : 10.24350/CIRM.V.20052503
Citer cette vidéo:
Krapp, Lothar Sebastian (2023). Definable convex and henselian valuations on ordered fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20052503
URI : http://dx.doi.org/10.24350/CIRM.V.20052503
|
Voir aussi
-
[Multi angle]
Beautiful pairs revisited
/ Auteur de la Conférence Ye, Jinhe.
-
[Multi angle]
An Imaginary Ax-Kochen/Ershov principle: the equicharacteristic zero case
/ Auteur de la Conférence Vicaria, Mariana.
-
[Multi angle]
Existential closedness of $\mathbb{Q}^{alg}$ as a globally valued field
/ Auteur de la Conférence Szachniewicz, Michal.
-
[Multi angle]
Interpretable, definably semisimple groups in various valued fields
/ Auteur de la Conférence Peterzil, Ya'acov.
-
[Multi angle]
The existential closedness problem for analytic solutions of difference equations
/ Auteur de la Conférence Padgett, Adele.
-
[Multi angle]
Multi topological fields and NTP2
/ Auteur de la Conférence Montenegro Guzman, Samaria.
-
[Multi angle]
A Hasse principle over Berkovich analytic curves
/ Auteur de la Conférence Mehmeti, Vlerë.
-
[Multi angle]
Beyond the Fontaine-Wintenberger theorem
/ Auteur de la Conférence Kartas, Konstantinos.
-
[Multi angle]
Around NIP Noetherian domains
/ Auteur de la Conférence Johnson, Will.
-
[Multi angle]
An invitation to globally valued fields
/ Auteur de la Conférence Hrushovski, Ehud.
-
[Multi angle]
Lang-Weil type bounds in finite difference fields
/ Auteur de la Conférence Hils, Martin.
-
[Multi angle]
Residue field domination
/ Auteur de la Conférence Haskell, Deirdre.
-
[Multi angle]
Motivic integration with pseudo-finite residue field and fundamental lemma
/ Auteur de la Conférence Forey, Arthur.
-
[Multi angle]
Tropical functions on skeletons
/ Auteur de la Conférence Ducros, Antoine.
-
[Multi angle]
Contracting endomorphisms of valued fields
/ Auteur de la Conférence Dor, Yuval.
-
[Multi angle]
Ax-Kochen-Ershov principles for finitely ramified henselian fields
/ Auteur de la Conférence Dittmann, Philip.
-
[Multi angle]
Group construction in $C$-minimal structures
/ Auteur de la Conférence Delon, Françoise.
-
[Multi angle]
Homology groups in algebraically closed valued fields
/ Auteur de la Conférence Cubides Kovacsics, Pablo.
-
[Multi angle]
Existential uniform p-adic integration and descent for integrability and largest poles
/ Auteur de la Conférence Cluckers, Raf.
-
[Multi angle]
Polynomials that vanish on many sets of codimension 2
/ Auteur de la Conférence Ben Yaacov, Itaï.
-
[Multi angle]
The model theory of Hardy fields and linear differential equations
/ Auteur de la Conférence Aschenbrenner, Matthias.
-
[Multi angle]
Transfer of decidability for existential theories of (valued) fields
/ Auteur de la Conférence Anscombe, Sylvy.
Bibliographie
- DITTMANN, Philip, JAHNKE, Franziska, KRAPP, Lothar Sebastian, et al. Definable valuations on ordered fields. arXiv preprint arXiv:2206.15301, 2022. - https://doi.org/10.48550/arXiv.2206.15301
- KRAPP, Lothar Sebastian, KUHLMANN, Salma, et LEHÉRICY, Gabriel. Ordered fields dense in their real closure and definable convex valuations. In : Forum Mathematicum. De Gruyter, 2021. p. 953-972. - https://doi.org/10.1515/forum-2020-0030
- KRAPP, Lothar Sebastian, KUHLMANN, Salma, et LEHÉRICY, Gabriel. Strongly NIP almost real closed fields. Mathematical Logic Quarterly, 2021, vol. 67, no 3, p. 321-328. - https://doi.org/10.1002/malq.202000060
- KRAPP, Lothar Sebastian, KUHLMANN, Salma, et LINK, Moritz. Definability of henselian valuations by conditions on the value group. The Journal of Symbolic Logic, 2022, p. 1-19. - https://doi.org/10.1017/jsl.2022.34