A note on existentially t-henselian fields

TL;DR

This paper explores the property of existentially t-henselian fields, showing its equivalence to $ ext{Z}$-largeness and examining its implications for the number of existential theories of henselian valuations.

Contribution

It provides new characterizations of existential t-henselianity and investigates the diversity of existential theories of henselian valuations on fields.

Findings

Existentially t-henselian fields are equivalent to $ ext{Z}$-large fields.

The property relates to whether $tF[ exttt{[}[t] exttt{]}]$ is Diophantine in $F( exttt{[}[t] exttt{]}])$.

The paper counts the number of existential theories of henselian valuations on a field.

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.