Model theory of tame valued fields and beyond: recent developments and open questions

2025/12/06 by Kuhlmann, Franz-Viktor
Mathematics · #12J10 #12L12 #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.2512.06386

openalex publication_date 2025/12/06 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

Abstract

We give a survey on recent developments in the model theory of valued fields since the introduction of the notion of ``tame valued field'', and of the modifications and generalizations of this notion.

Citations

• On certain definable coarsenings of valuation rings and their applications
• On Lambda functions in henselian and separably tame valued fields
• Definable henselian valuations in positive residue characteristic
• Kähler differentials of extensions of valuation rings and deeply ramified fields
• Ax-Kochen-Ershov principles for finitely ramified henselian fields
• Beyond the Fontaine-Wintenberger theorem
• Defect extensions and a characterization of tame fields
• Diophantine problems over tamely ramified fields
• The valuation theory of deeply ramified fields and its connection with defect extensions
• The valuation theory of deeply ramified fields and its connection with defect extensions
• On valuation independence and defectless extensions of valued fields
• The algebra and model theory of tame valued fields
• Perfectoid spaces
• Diophantine Problems Over Local Fields I

Related

• The algebra and model theory of tame valued fields
• Model theory of valued fields
• Model Theory: Combinatorics, Groups, Valued Fields and Neostability
• The model theory of separably tame valued fields
• A note on valued fields with finite group actions