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A note on existentially t-henselian fields
Sylvy Anscombe
https://arxiv.org/abs/2603.27612 https://arxiv.org/pdf/2603.27612 https://arxiv.org/html/2603.27612
arXiv:2603.27612v1 Announce Type: new
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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