Related papers: Definable henselian valuations on dp-minimal real …

We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero.

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…

We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of…

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef's paper [Invent. Math, 77 (1984)]. We derive from it the…

We are concerned with topology of Hensel minimal structures on non-trivially valued fields $K$, whose axiomatic theory was introduced in a recent paper by Cluckers-Halupczok-Rideau. We additionally require that every definable subset in the…

We show that every definable nested family of closed and bounded subsets of a $P$-minimal field $K$ has non-empty intersection. As an application we answer a question of Darni\`ere and Halupczok showing that $P$-minimal fields satisfy the…

We prove the existence of definable retractions onto arbitrary closed subsets of $K^{n}$ definable over Henselian valued fields $K$. Hence directly follows non-Archimedian analogues of the Tietze--Urysohn and Dugundji theorems on extending…

We continue our earlier investigation of dp-finite fields. We show that the "heavy sets" of [6] are exactly the sets of full dp-rank. As a consequence, full dp-rank is a definable property in definable families of sets. If $I$ is the group…

Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ is one of the following: (i)…

Let $K$ be a Henselian, non-trivially valued field with separated analytic structure. We prove the existence of definable retractions onto an arbitrary closed definable subset of $K^{n}$. Hence directly follow definable non-Archimedean…

In this note we show that groups with definable generics in a separably closed valued of finite imperfection degree can be embedded into groups definable in their algebraic closure.

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 classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from…

We prove that every ultraproduct of $p$-adics is inp-minimal (i.e., of burden $1$). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic $0$ in the RV…

So far there exist just a few results about the uniqueness of maximal immediate valued differential field extensions and about the relationship between differential-algebraic maximality and differential-henselianity; see arXiv:1509.02588,…

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is $0$-hmix-minimality (which, in…

We give a complete list of the one-dimensional groups definable in algebraically closed valued fields and i the pseudo-local fields, up to a finite index subgroup and a quotient by a finite subgroup.

We show that asymptotic (valued differential) fields have unique maximal immediate extensions. Connecting this to differential-henselianity, we prove that any differential-henselian asymptotic field is differential-algebraically maximal,…

We study the class of differentially henselian fields, which are henselian valued fields equipped with generic derivations in the sense of Cubides Kovacics and Point, and are special cases of differentially large fields in the sense of…

We study the behaviour of forking in valued fields, and we give several sufficient conditions for parameter sets in a Henselian valued field of residue characteristic zero to be an extension base. Notably, we consider arbitrary (potentially…