The Ax-Kochen-Ershov principles via the higher valued hyperfield
Junguk Lee
TL;DR
The paper develops Ax-Kochen-Ershov principles for finitely ramified henselian valued fields by translating their model-theoretic behavior into higher valued hyperfields $\mathcal{H}_{\nu,n}$. It introduces a robust framework combining strict Cohen subrings, valued hyperfields, and a lifting mechanism that converts hyperfield homomorphisms into valued-field embeddings, enabling transfers of elementary and existential theories. The main results establish relative completeness, existential completeness, and existential closedness via hyperfields, yielding decidability transfers for full and existential theories and unifying prior work by Basarab, Lee, and Anscombe–Dittmann–Jahnke. This hyperfield-centric approach provides new tools for analyzing valued fields and highlights a pathway to automate and compare theories through hyperfield invariants, with potential broad impact on the model theory of valued fields. The work thereby extends the Ax-Kochen-Ershov paradigm to a broader class of valued fields without resorting to perfect residue fields, incorporating tame and wild ramification in a unified hyperfield setting.
Abstract
In this paper, we concern the model theory of finitely ramified henselian valued fields via higher valued hyperfields. Most of all, we provide a number of Ax-Kochen-Ershov Theorems for finitely ramified henselian valued fields relative to higher valued hyperfields. As corollaries, we deduce a transfer of decidability for full theories and existential theories of a finitely ramified henselian valued fields relative to higher valued hyperfields.