On logical characterization of henselianity

TL;DR

This paper explores conditions under which valued fields with certain logical properties, like quantifier elimination, are guaranteed to be henselian, especially focusing on fields with specific value groups and language frameworks.

Contribution

It establishes new sufficient conditions linking quantifier elimination in specific languages to henselianity in valued fields, broadening understanding of logical characterizations.

Findings

Valued fields with quantifier elimination in the Macintyre language are henselian.

Fields with a Z-group as value group and quantifier elimination in the Denef-Pas language are henselian.

A large class of Denef-Pas style languages naturally relate to henselianity.

Abstract

We give some sufficient conditions under which any valued field that admits quantifier elimination in the Macintyre language is henselian. Then, without extra assumptions, we prove that if a valued field of characteristic $(0,0)$ has a $\Z$-group as its value group and admits quantifier elimination in the main sort of the Denef-Pas style language $\mathcal{L}_{RRP}$ then it is henselian. In fact the proof of this suggests that a quite large class of Denef-Pas style languages is natural with respect to henselianity.