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

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.