A definable henselian valuation with high quantifier complexity

TL;DR

The paper constructs a parameter-free definable henselian valuation ring that cannot be defined by simple quantifier patterns, answering a question posed by Prestel.

Contribution

It provides the first example of a henselian valuation with high quantifier complexity that is definable without parameters.

Findings

Existence of a parameter-free definable henselian valuation with complex quantifier structure

Answer to Prestel's question on definability patterns

Illustration of limitations in simple definability formulas

Abstract

We give an example of a parameter-free definable henselian valuation ring which is neither definable by a parameter-free $\forall\exists$-formula nor by a parameter-free $\exists\forall$-formula in the language of rings. This answers a question of Prestel.