On the quantifier complexity of definable canonical henselian valuations

TL;DR

This paper investigates the definability of the canonical henselian valuation in fields, showing that in most cases it can be defined by simple logical formulas such as universal-existential or existential-universal types.

Contribution

It demonstrates that the canonical henselian valuation is often definable by simple logical formulas, advancing understanding of valuation definability in field theory.

Findings

Most canonical henselian valuations are definable by universal-existential formulas.

In many cases, these valuations are also definable by existential-universal formulas.

The work clarifies the logical complexity of defining canonical henselian valuations.

Abstract

We discuss definability in the language of rings without parameters of the unique canonical henselian valuation of a field. We show that in most cases where the canonical henselian valuation is definable, it is already definable by a universal-existential or an existential-universal formula.