Related papers: Definable transformation to normal crossings over …
The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is…
We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach…
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…
Let $K$ be a complete non-Archimedean field $K$ with separated power series, treated in the analytic Denef--Pas language. We prove the existence of definable retractions onto an arbitrary closed definable subset of $K^{n}$, whereby…
The paper concerns uniform Yomdin-Gromov parametrizations together with an estimate of their number, which generalizes a theorem by Cluckers-Forey-Loeser to arbitrary equicharacteristic zero valued fields with analytic structure. To this…
We develop a real-analytic framework, called perplex analysis, in which the complex, split-complex, and dual numbers arise as members of a single four-parameter family of two-dimensional commutative real algebras. Within this unified…
In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to…
We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries, [DD], on $p$-adic subanalytic sets, and we continue the study of non-archimedean fields with…
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…
This paper develops algebraic geometry over Henselian real valued (i.e. of rank 1) fields $K$, being a sequel to our paper about that over Henselian discretely valued fields. Several results are given including: a certain concept of fiber…
We develop geometry of affine algebraic varieties in $K^{n}$ over Henselian rank one valued fields $K$ of equicharacteristic zero. Several results are provided including: the projection $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ and…
Quantization, at least in some formulations, involves replacing some algebra of observables by a (more non-commutative) deformed algebra. In view of the fundamental role played by K-theory in non-commutative geometry and topology, it is of…
We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure,…
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…
Global constructions of quantization deformation and obstructions are discussed for an arbitrary complex analytic space in terms of adapted (analytic) Hochschild cohomology. For K3-surfaces an explicit global construction of a Poisson…
Let $T$ be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has the exchange…
In this thesis we investigate a new formalism for supergeometry which focuses on the categorical properties of the theory. This approach is our main tool in the subsequent investigation of a global analytic approach to the construction of…
The main purpose of the paper is to establish a closedness theorem over Henselian valued fields $K$ of equicharacteristic zero (not necessarily algebraically closed) with separated analytic structure. It says that every projection with a…
We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group.…
Generalized analytic functions over generalized analytic manifolds are build from sums of convergent real power series with non-negative real exponents (and some well-ordering condition on the support). In a paper by Mart\'in-Villaverde,…