Differentially Henselian Fields
Gabriel Ng
TL;DR
This work introduces and develops the theory of differential henselian fields, defined as henselian valued fields endowed with a derivation satisfying the axiom scheme $(\mathrm{DL})$ and shown to be equivalent to Guzy–Point’s generic-derivation framework $\mathrm{UC}'_1$. Building a bridge from classical henselian valued fields to valued-differential contexts, the authors establish Ax–Kochen/Ershov lifting results, relative embedding theorems, and relative quantifier elimination in the Pas language for differential $\mathrm{ac}$-valued fields, together with stable embeddedness results for value groups and residue fields. They provide constructive production methods via iterated Hahn-series extensions and transcendence-basis arguments, and prove robustness under differential Weil descent, including preservation under algebraic extensions and connection to differential-closed valued fields. By unifying lifting principles, QE, and descent in the differential setting, the paper broadens the toolkit for model-theoretic analysis of valued-differential fields and paves the way for concrete realizations of differentially henselian fields in both pure theory and explicit constructions.
Abstract
We study the class of differentially henselian fields, which are henselian valued fields equipped with generic derivations in the sense of Cubides Kovacics and Point, and are special cases of differentially large fields in the sense of León Sánchez and Tressl. We prove that many results from henselian valued fields as well as differentially large fields can be lifted to the differentially henselian setting, for instance Ax-Kochen/Ershov principles, characterisations in terms of differential algebras, etc. We also give methods to concretely construct such fields in terms of iterated power series expansions and inductive constructions on transcendence bases.