| Résume | I will discuss several open problems for valued fields of positive residue characteristic. Many of them are mentioned in the survey paper [1], but I will provide
more background and give some updates. Some main topics will be:
1) Given a valued field, some of its maximal immediate extensions may have better model theoretic properties than others. For instance, this can be observed
in [2]. If the valued field has a truncation closed embedding in a power series field, this may extend to some of its maximal immediate extensions, but not
to others. Is there a connection between the two problems? What does the existence of truncation closed embeddings say about model theoretic properties?
I will introduce the notion of extremal valued field (see [3]). Is every such field existentially closed in all (or some) of its maximal immediate extensions? I will
also list other problems about extremal valued fields. It is important to investigate them, because for instance Fp((t)) is extremal.
2) Under which additional conditions is a valued field K existentially closed in a function field F that admits a K-rational place P ? Does this always hold after a
finite constant extension K′|K? What can be said about the relation of P to K-rational discrete places of F ? What can be said if P admits local uniformization?
For background, see [4].
3) Is a valued field roughly deeply ramified if all of its extensions are independent defect fields? Definitions and background from [5] will be given in my talk.
4) The Henselian Rationality Theorem (see [6]) works over tame fields. To which extent does it also work over perfect fields, deeply ramified fields, or extremal
fields of positive characteristic?
[1] Kuhlmann, F.-V.: Model theory of tame valued fields and beyond: recent developments and open questions, submitted; arXiv:2512.06386
[2] Kartas, K.: Decidability via the tilting correspondence, Algebra and Number Theory 18 (2024), 209–24
[3] Anscombe, S. – Kuhlmann, F.-V.: Notes on extremal and tame valued fields, J. Symb. Logic 81 (2016), 400–416
[4] Kuhlmann, F.-V.: On places of algebraic function fields in arbitrary characteristic, Advances in Math. 188 (2004), 399–424
[5] Kuhlmann, F.-V. – Rzepka, A.: The valuation theory of deeply ramified fields and its connection with defect extensions, Transactions Amer. Math. Soc. 376
(2023), 2693–2738
[6] Kuhlmann, F.-V.: Elimination of Ramification II: Henselian Rationality, Israel J. Math. 234 (2019), 927–958 |
