Arithmetic of fields: model theory, valuations, and geometry / L'arithmétiques des corps : théories des modèles, valuations, et geometrie

Collection Arithmetic of fields: model theory, valuations, and geometry / L'arithmétiques des corps : théories des modèles, valuations, et geometrie

Organizer(s) Anscombe, Sylvy ; Bagayoko, Vincent ; Forey, Arthur ; Jahnke, Franziska

Date(s) 06/04/2026 - 10/04/2026

linked URL https://conferences.cirm-math.fr/3500.html

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Order-topological characterization of groups and fields with automatic $\emptyset$-definability

An ordered abelian group G (resp. a field k) has automatic (0)-definability if, in every henselian valued field with value group G (resp. residue field k), the valuation is (0)-definable in the language of rings. In joint work with B. Boissonneau, F. Jahnke, and P. Touchard, we characterize such groups via the model-theoretic properties of weak and strong augmentability by infinitesimals, and such fields via t-henselianity, in characteristic 0. In this talk, I show that strong augmentability admits an order-topological formulation, and I discuss analogous questions for weak augmentability and t-henselianity in the setting of orderable fields. The talk includes results from this joint work, as well as ongoing work with L. S. Krapp and S. Kuhlmann.

Information about the video

Date of recording 06/04/2026
Date of publication 30/04/2026
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.20467103
Cite this video De Mase, Anna (06/04/2026). Order-topological characterization of groups and fields with automatic $\emptyset$-definability. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20467103
URL https://dx.doi.org/10.24350/CIRM.V.20467103

Domain(s)

Logic

Bibliography

BOISSONNEAU, Blaise, DE MASE, Anna, JAHNKE, Franziska, et al. Growing Spines: Ad Infinitum et Ad Infinitesimalia. arXiv preprint arXiv:2512.04932, 2025. - https://doi.org/10.48550/arXiv.2512.04932
KRAPP, Lothar Sebastian, KUHLMANN, Salma, et LINK, Moritz. Definability of henselian valuations by conditions on the value group. The Journal of Symbolic Logic, 2023, vol. 88, no 3, p. 1064-1082. - https://doi.org/10.1017/jsl.2022.34
PRESTEL, Alexander. Algebraic number fields elementarily determined by their absolute Galois group. Israel Journal of Mathematics, 1991, vol. 73, no 2, p. 199-205. - https://doi.org/10.1007/BF02772949
PRESTEL, Alexander et ZIEGLER, Martin. Model theoretic methods in the theory of topological fields. Journal für die reine und angewandte Mathematik, 1978, vol. 302, p. 318-341. - https://doi.org/10.1515/crll.1978.299-300.318