Definable V-topologies, Henselianity and NIP

Author:

Halevi Yatir1,Hasson Assaf1,Jahnke Franziska2

Affiliation:

1. Department of Mathematics, Ben Gurion University of the Negev, Be’er Sehva, Israel

2. Westfälische Wilhelms-Universität Münster, Institut für Mathematische Logik und Grundlagenforschung, Einsteinstr 62, 48149 Münster, Germany

Abstract

We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian (respectively, [Formula: see text]-henselian), then [Formula: see text] and [Formula: see text] are comparable (respectively, dependent). As a consequence, Shelah’s conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah’s conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is [Formula: see text]-henselian.

Funder

European Research Council

ISF

Deutsche Forschungsgemeinschaft

Publisher

World Scientific Pub Co Pte Lt

Subject

Logic

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1. Characterizing NIP henselian fields;Journal of the London Mathematical Society;2024-03

2. FINITE UNDECIDABILITY IN NIP FIELDS;The Journal of Symbolic Logic;2023-10-04

3. Definable valuations on ordered fields;Model Theory;2023-06-26

4. The étale-open topology and the stable fields conjecture;Journal of the European Mathematical Society;2023-06-14

5. Henselian expansions of NIP fields;Journal of Mathematical Logic;2023-04-29

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