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PreprintCC BY 4.0arXiv

AKE principles in roughly deeply ramified henselian valued fields

  • Franziska Jahnke
  • Margarete Ketelsen
  • Floris Vermeulen
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Abstract

We show that for any henselian valued field of mixed characteristic (0,p), the (existential) β„’val-theory of the valued field is determined by the (existential) theory of the value group in β„’oag with a constant for v(p) and the (existential) theory of the residue ring π’ͺv/(p) in an expansion β„’Witt of the language of rings, provided π’ͺv/(p) is semi perfect. We moreover show that the β„’Witt-structure on π’ͺv/(p) is β„’ring-definable using constants, and that this is exactly the structure induced on π’ͺv/(p) by the ambient valued field. As a consequence, we obtain a relative quantifier elimination result (eliminating K-quantifiers) in a suitable language for the theory of roughly deeply ramified henselian valued fields of mixed characteristic (0,p).

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Cite

BibTeXRIS
BibTeX
@misc{jahnke2026principles,
  title = {{AKE principles in roughly deeply ramified henselian valued fields}},
  author = {Franziska Jahnke and Margarete Ketelsen and Floris Vermeulen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15376},
  eprint = {2609.15376},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15376},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - AKE principles in roughly deeply ramified henselian valued fields
AU  - Franziska Jahnke
AU  - Margarete Ketelsen
AU  - Floris Vermeulen
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15376
UR  - https://arxiv.org/abs/2609.15376
AB  - We show that for any henselian valued field of mixed characteristic (0,p), the (existential) β„’_val-theory of the valued field is determined by the (existential) theory of the value group in β„’_oag with a constant for v(p) and the (existential) theory of the residue ring π’ͺ_v/(p) in an expansion β„’_Witt of the language of rings, provided π’ͺ_v/(p) is semi perfect. We moreover show that the β„’_Witt-structure on π’ͺ_v/(p) is β„’_ring-definable using constants, and that this is exactly the structure induced on π’ͺ_v/(p) by the ambient valued field. As a consequence, we obtain a relative quantifier elimination result (eliminating K-quantifiers) in a suitable language for the theory of roughly deeply ramified henselian valued fields of mixed characteristic (0,p).
ER  -