On Lambda functions in henselian and separably tame valued fields

TL;DR

This paper characterizes Lambda closures in henselian and separably tame valued fields, providing a new language-based description, and explores their applications in definability and model theory, including a form of local quantifier elimination.

Contribution

It introduces a simple, generator-based description of Lambda closures using expanded language with Lambda functions and extends the theory of separably tame valued fields to more general cases.

Findings

Lambda closures are generated by Lambda functions applied to generators.

Existentially definable sets can be locally parameterized by large sets with nonempty topology.

Extended theory supports embedding theorems and Ax--Kochen/Ershov principles.

Abstract

Given a field extension $F/C$, the ``Lambda closure'' $\Lambda_{F}C$ of $C$ in $F$ is a subextension of $F/C$ that is minimal with respect to inclusion such that $F/\Lambda_{F}C$ is separable. The existence and uniqueness of $\Lambda_{F}C$ was proved by Deveney and Mordeson in 1977. We show that it admits a simple description in terms of given generators for $C$: we expand the language of rings by the parameterized Lambda functions, and then $\Lambda_{F}C$ is the subfield of $F$ generated over $C$ by additionally closing under these functions. We then show that, given particular generators of $C$, $\Lambda_{F}C$ is the subfield of $F$ generated iteratively by the images of the generators under Lambda functions taken with respect to $p$-independent tuples also drawn from those generators. We apply these results to given a ``local description'' of existentially definable sets in fields equipped with a henselian topology. Let $X(K)$ be an existentially definable set in the theory of a field $K$ equipped with a henselian topology $\tau$. We show that there is a definable injection into $X(K)$ from a Zariski-open subset $U_{1}^{\circ}$ of a set with nonempty $\tau$-interior, and that each element of $U_{1}^{\circ}$ is interalgebraic (over parameters) with its image in $X(K)$. This can be seen as a kind of {\em very weak local quantifier elimination}, and it shows that existentially definable sets are (at least generically and locally) definably pararameterized by ``big'' sets. In the final section we extend the theory of separably tame valued fields, developed by Kuhlmann and Pal, to include the case of infinite degree of imperfection, and to allow expansions of the residue field and value group structures. We prove an embedding theorem which allows us to deduce the usual kinds of resplendent Ax--Kochen/Ershov principles.