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Definable valuations on ordered fields

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2023

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Dittmann, Philip
Jahnke, Franziska

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https://doi.org/10.2140/mt.2023.2.101
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Deutsche Forschungsgemeinschaft (DFG): EXC 2044-390685587

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Model Theory. Mathematical Sciences Publishers MSP. 2023, 2(1), S. 101-120. ISSN 2832-904X. eISSN 2832-9058. Verfügbar unter: doi: 10.2140/mt.2023.2.101

Zusammenfassung

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 Lr and in the richer language of ordered rings Lor. 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 Lor but not in the language Lr, any Lor-definable henselian valuation is already Lr-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 and an ordered field, respectively). Moreover, we show that in almost real closed fields any Lor-definable valuation is henselian.

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Fachgebiet (DDC)
510 Mathematik

Schlagwörter

definable valuations, ordered fields, convex valuations, henselian valuations, stably embedded, almost real closed fields

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ISO 690DITTMANN, Philip, Franziska JAHNKE, Lothar Sebastian KRAPP, Salma KUHLMANN, 2023. Definable valuations on ordered fields. In: Model Theory. Mathematical Sciences Publishers MSP. 2023, 2(1), S. 101-120. ISSN 2832-904X. eISSN 2832-9058. Verfügbar unter: doi: 10.2140/mt.2023.2.101
BibTex
@article{Dittmann2023-06-26Defin-75455,
  title={Definable valuations on ordered fields},
  year={2023},
  doi={10.2140/mt.2023.2.101},
  number={1},
  volume={2},
  issn={2832-904X},
  journal={Model Theory},
  pages={101--120},
  author={Dittmann, Philip and Jahnke, Franziska and Krapp, Lothar Sebastian and Kuhlmann, Salma}
}
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