Related papers: Henselianity in the language of rings
We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative…
We discuss definability in the language of rings without parameters of the unique canonical henselian valuation of a field. We show that in most cases where the canonical henselian valuation is definable, it is already definable by a…
We give model theoretic criteria for $\exists \forall$ and $\forall \exists$- formulas in the ring language to define uniformly the valuation rings $\mathcal{O}$ of models $(K, \mathcal{O})$ of an elementary theory $\Sigma$ of henselian…
We study the question of $\mathcal{L}_{\mathrm{ring}}$-definability of non-trivial henselian valuation rings. Building on previous work of Jahnke and Koenigsmann, we provide a characterization of henselian fields that admit a non-trivial…
Admitting a non-trivial $p$-henselian valuation is a weaker assumption on a field than admitting a non-trivial henselian valuation. Unlike henselianity, $p$-henselianity is an elementary property in the language of rings. We are interested…
We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue…
We study the definability of convex valuations on ordered fields, with a particular focus on the distinguished subclass of henselian valuations. In the setting of ordered fields, one can consider definability both in the language of rings…
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,…
We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian…
We introduce a notion of valued module which is suitable to study valued fields of positive characteristic. Then we built-up a robust theory of henselianity in the language of valued modules and prove Ax-Kochen Ershov type results.
We study the question which henselian fields admit definable henselian valuations (with or without parameters). We show that every field which admits a henselian valuation with non-divisible value group admits a parameter-definable…
We investigate the model completeness of the theory of a mixed characteristic henselian valued field with finite ramification relative to the residue field and value group. We address the case in which the valued field has a value group…
We investigate what henselian valuations on ordered fields are definable in the language of ordered rings. This leads towards a systematic study of the class of ordered fields which are dense in their real closure. Some results have…
Given a henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any henselian valuation whose value group is not closed in its divisible hull is definable in the…
We show that every henselian valued field $L$ of residue characteristic 0 admits a proper subfield $K$ which is dense in $L$. We present conditions under which this can be taken such that $L|K$ is transcendental and $K$ is henselian. These…
In this note we investigate the question whether a henselian valued field carries a non-trivial 0-definable henselian valuation (in the language of rings). It follows from the work of Prestel and Ziegler that there are henselian valued…
Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years. We survey these results which address the definability of…
For an arbitrary valued field $(K,v)$ and a given extension $v(K^*)\hookrightarrow\Lambda$ of ordered groups, we analyze the structure of the tree formed by all $\Lambda$-valued extensions of $v$ to the polynomial ring $K[x]$. As an…
In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the…
We give an example of a parameter-free definable henselian valuation ring which is neither definable by a parameter-free $\forall\exists$-formula nor by a parameter-free $\exists\forall$-formula in the language of rings. This answers a…