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arXiv · 2601.17395

Existential fragments of theories of henselian valued fields

Sylvy Anscombe·Arno Fehm
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Abstract

We study fragments of the existential theory of henselian valued fields with parameters. This includes the $\exists_n$-fragment in the equicharacteristic or unramified mixed characteristic case, the $\exists_n\exists_1$-fragment in the equicharacteristic case, and the $\exists_n$-fragment in the residue characteristic zero case. For example, we obtain an unconditional axiomatization (and thereby decidability) of the $\exists_3$-theory of $\mathbb{F}_{q}(\!(t)\!)$ in the language of valued fields with a parameter for $t$.

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Sylvy Anscombe, Arno Fehm. 2026-01-24. Existential fragments of theories of henselian valued fields. https://arxiv.org/abs/2601.17395

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