Construction of a valued field whose valuation ring is definable but neither $\exists\forall\exists$ nor $ \forall\exists\forall$-definable in the language of rings

TL;DR

This paper constructs a valued field with a definable valuation ring that cannot be defined by certain restricted logical formulas, highlighting limitations in definability within the language of rings.

Contribution

It provides the first example of a valued field where the valuation ring is definable but not by any $orall ext{-}orall ext{-}orall$ or $orall ext{-}orall ext{-}orall$-formulas, demonstrating definability limitations.

Findings

Valuation ring is definable without parameters in the language of rings.

No $orall ext{-}orall ext{-}orall$ or $orall ext{-}orall ext{-}orall$-formula defines the valuation ring.

The example shows restrictions on logical formulas for defining valuation rings.

Abstract

We give an example of a valued field $(K,A)$ such that the valuation ring $A$ is definable by an $L_{\text{ring}}$-formula without parameters, but there is no $\exists\forall\exists$ or $\forall\exists\forall$-formula in $L_{\text{ring}}$ to define it.