On logical characterization of 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 $\mathbb{Z}$-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.

Minor corrections