Relative Quantifier Elimination for Separable-Algebraically Maximal Kaplansky Fields

TL;DR

This paper proves quantifier elimination results for a class of valued fields called separable-algebraically maximal Kaplansky fields, leading to model-theoretic simplifications and applications to NIP fields.

Contribution

It establishes quantifier elimination in a three-sorted language for these fields, extending the AKE principles and analyzing their model-theoretic properties.

Findings

Quantifier elimination down to residue field and value group.

Resplendent elimination of field quantifiers in NIP and NIP$_n$ henselian fields.

Reduction of existential formulas to quantifier-free formulas in the home sort.

Abstract

Let $C$ be the class of separable-algebraically maximal equi-characteristic Kaplansky fields of a given imperfection degree, admitting an angular component map. We prove that the common theory of the class $C$ resplendently eliminates quantifiers down to the residue field and the value group, in a three sorted language of valued fields with a symbol for an angular component map and symbols for the parameterized lambda-functions. As a consequence, we obtain that equi-characteristic NIP and NIP$_n$ henselian fields with an angular component map resplendently eliminate field quantifiers in this language. We also prove that this elimination reduces existential formulas to existential formulas without quantifiers from the home sort. Finally, we draw several conclusions following the AKE philosophy for elements of the class $C$, including the usual AKE principles for $\equiv,\equiv_{\exists},\preceq,\preceq_{\exists}$ and for relative decidability.