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arXiv · 2508.06876

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

Mohsen Khani·Shaghayegh Shirani·Zahra Yadegari·Afshin Zarei
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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.

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Mohsen Khani, Shaghayegh Shirani, Zahra Yadegari, Afshin Zarei. 2025-08-09. 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. https://arxiv.org/abs/2508.06876

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