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arXiv · 2603.27612

A note on existentially t-henselian fields

Sylvy Anscombe
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Abstract

A field is existentially t-henselian if it is has the same existential theory in the first-order language of rings as a field that admits a nontrivial henselian valuation. This property turns out to be equivalent to $\mathbb{Z}$-largeness, which is a property identified in previous work with Fehm, and which holds for $F$ if and only if $tF[\![t]\!]$ is not Diophantine in $F(\!(t)\!)$, without extra constants. In this short note, we further investigate this property in order to count the number of existential theories of henselian valuations on a given field, and to find other characterizations of existential t-henselianity.

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Sylvy Anscombe. 2026-03-29. A note on existentially t-henselian fields. https://arxiv.org/abs/2603.27612

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