Eliminating Field Quantifiers in Strongly Dependent Henselian Fields

TL;DR

This paper proves that in strongly dependent henselian fields, all field quantifiers can be eliminated in the Denef-Pas language, extending to their henselizations, with implications for model theory and valued fields.

Contribution

It establishes quantifier elimination for strongly dependent henselian fields and generalizes the result to a broader class of fields, including algebraically maximal Kaplansky fields.

Findings

Quantifier elimination in the Denef-Pas language for strongly dependent henselian fields.

Strong dependence is preserved under henselization.

Extension of results to algebraically maximal Kaplansky fields.

Abstract

We prove elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if $(K,v)$ is strongly dependent then so is its henselization.