arXiv subjects

Krzysztof Jan Nowak

Publications and source records attributed to Krzysztof Jan Nowak.

At least 19 recordsLinked to original sources

arXiv2026

Extension of Lipschitz maps definable in Hensel minimal structures

In this paper, we establish a theorem on extension of Lipschitz maps $f$ definable in Hensel minimal fields $K$. This may be regarded as a definable, non-Archimedean, non-locally compact version of Kirszbraun's extension theorem. We proceed with double induction with respect to the dimensions of the ambient space and of the domain of $f$. To this end, we introduce the concept of a definable open cell package with a skeleton which, along with the concept of a risometry, plays a key role in our induction procedure.

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arXiv2026

Strong stratifications and uniform Yomdin-Gromov parametrizations in valued fields with analytic structure

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 end, we establish a certain strong analytic stratification of definable sets, based on a definable non-Archimedean version of Bierstone-Milman's canonical desingularization algorithm from our earlier paper. Other basic tools are: elimination of valued field quantifiers, term description of functions definable in analytic structures, and Lipschitz cell decomposition compatible with $RV$-parametrized sets in Hensel minimal structures.

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arXiv2026

About closed subsets definable in Hensel minimal structures

The main purpose is to establish two theorems about closed 0-definable subsets $A$ of an affine space $K^{n}$ over a Hensel minimal field $K$. The first, being a non-Archimedean counterpart of one from o-minimal geometry, states that every such subset $A$ is the zero locus of a continuous 0-definable function on $K^{n}$. The second is a definable, non-Archimedean version of the Tietze-Urysohn extension theorem. The proofs use ubiquity of clopen sets in non-Archimedean geometry and a description of definable sets.

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arXiv2024

Tame topology in Hensel minimal structures

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 imaginary sort $RV$, binding together the residue field $Kv$ and value group $vK$, be already definable in the plain valued field language. This condition is satisfied by several classical tame structures on Henselian fields, including Henselian fields with analytic structure, V-minimal fields, and polynomially bounded o-minimal structures with a convex subring. In this article, we establish many results concerning definable functions and sets; among others, existence of the limit for definable functions of one variable, a closedness theorem, several non-Archimedean versions of the Lojasiewicz inequalities, an embedding theorem for regular definable spaces, and the definable ultranormality and ultraparacompactness of definable Hausdorff LC-spaces.

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arXiv2024

Pulling back singularities for analytic complete intersections

The following pullback problem will be considered. Given a finite holomorphic map germ $ϕ: (\mathbb{C}^{n}, 0) \to (\mathbb{C}^{n}, 0)$ and an analytic germ $X$ in the target, if the preimage $Y = ϕ^{-1}(X)$, taken with the reduced structure, is smooth, so is $X$. The main aim of this paper is to give an affirmative solution for $X$ being a geometric complete intersection. The case, where $Y$ is not contained in the ramification divisor $Z$ of $ϕ$, was established by Ebenfelt-Rothschild (2007) and afterwards by Lebl (2008) and Denkowski (2016). The hypersurface case was achieved by Giraldo-Roeder (2020) and recently by Jelonek (2023).

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arXiv2020

A closedness theorem and applications in geometry of rational points over Henselian valued fields

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 is again the closedness theorem that the projections $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, Hölder continuity of definable functions on closed bounded subsets of $K^{n}$, the existence of definable retractions onto closed definable subsets of $K^{n}$, and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent preprint, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with some applications.

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arXiv2019

Definable transformation to normal crossings over Henselian fields with separated analytic structure

We are concerned with rigid analytic geometry in the general setting of Henselian fields $K$ with separated analytic structure, whose theory was developed by Cluckers--Lipshitz--Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone--Milman so that both these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to $K$. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions.

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arXiv2019

Definable retractions and a non-Archimedean Tietze--Urysohn theorem over Henselian valued fields

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 continuous definable functions. The main ingredients of the proof are a description of definable sets due to van den Dries, resolution of singularities and our closedness theorem.

math.AG↗

arXiv2019

Definable retractions over complete fields with separated power series

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 definable non-Archimedean versions of the extension theorems by Tietze--Urysohn and Dugundji follow directly. We reduce the problem to the case of a simple normal crossing divisor, relying on our closedness theorem and desingularization of terms. The latter result is established by means of the following tools: elimination of valued field quantifiers (due to Cluckers--Lipshitz--Robinson), embedded resolution of singularities by blowing up (due to Bierstone--Milman or Temkin), the technique of quasi-rational subdomains (due to Lipshitz--Robinson) and our closedness theorem.

math.AG↗

arXiv2019

Definable retractions over Henselian valued fields with analytic structure

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 versions of the extension theorems by Tietze--Urysohn and Dugundji. This generalizes our previous paper dealing with complete non-Archimedean fields with separated power series and remains true for Henselian valued fields with strictly convergent analytic structure, because every such a structure can be extended in a definitional way to a separated analytic structure. Our proof uses a variant of the one from that paper, based on canonical resolution of singularities, and a model-theoretic compactness argument.

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arXiv2018

Some results of geometry over Henselian fields with analytic structure

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 a complete rank one valued fields with the Tate algebra of strictly convergent power series. The algebraic case was treated in our previous papers. Here we are going to carry over the research to the general analytic settings. Also considered are the local rings of analytic function germs induced by a given separated Weierstrass system, which turn out to be excellent and regular. Several results are established as, for instance, piecewise continuity of definable functions, several versions of the Łojasiewicz inequality, Hölder continuity of definable functions continuous on closed bounded subsets of the affine space and curve selection for definable sets. Likewise as before, at the center of our approach is the closedness theorem to the effect that every projection with closed bounded fiber is a definably closed map. It enables application of resolution of singularities and of transformation to a normal crossing by blowing up (here applied to the local rings of analytic function germs) in much the same way as over locally compact ground fields. We rely on elimination of valued field quantifiers, term structure of definable functions and b-minimal cell decomposition, due to Cluckers--Lipshitz--Robinson, as well as on relative quantifier elimination for ordered abelian groups, due to Cluckers--Halupczok. Besides, other two ingredients of the proof of the closedness theorem are existence of the limit (after finite partitioning of the domain) for a definable function of one variable and fiber shrinking, being a relaxed version of curve selection.

math.AG↗

arXiv2018

A closedness theorem over Henselian valued fields with analytic structure

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 projective fiber is a definably closed map. This remains valid also for valued fields with analytic structure induced by a strictly convergent Weierstrass systems, including the classical, complete rank one valued fields with the Tate algebra of strictly convergent power series. As application, we prove two theorems on existence of the limit and on piecewise continuity.

math.AG↗

arXiv2017

The closedness theorem over Henselian valued fields

We prove the closedness theorem over Henselian valued fields, which was established over rank one valued fields in one of our recent papers. In the proof, as before, we use the local behaviour of definable functions of one variable and the so-called fiber shrinking, which is a relaxed version of curve selection. Now our approach applies also relative quantifier elimination for ordered abelian groups due to Cluckers--Halupczok. Afterwards the closedness theorem will allow us to achieve i.a. the Łojasiewicz inequality, curve selection and extending hereditarily rational functions as well as to develop the theory of regulous functions and sheaves.

math.AG↗

arXiv2017

On functions given by algebraic power series over Henselian valued fields

This paper provides, over Henselian valued fields, some theorems on implicit function and of Artin--Mazur on algebraic power series. Also discussed are certain versions of the theorems of Abhyankar--Jung and Newton--Puiseux. The latter is used in analysis of functions of one variable, definable in the language of Denef--Pas, to obtain a theorem on existence of the limit, proven over rank one valued fields in one of our recent papers. This result along with the technique of fiber shrinking (developed there over rank one valued fields) were, in turn, two basic tools in the proof of the closedness theorem.

math.AG↗

arXiv2017

Piecewise continuity of functions definable over Henselian rank one valued fields

Consider a Henselian rank one valued field $K$ of equicharacteristic zero along with the language $\mathcal{L}^{P}$ of Denef--Pas. Let $f: A \to K$ be an $\mathcal{L}^{P}$-definable (with parameters) function on a subset $A$ of $K^{n}$. We prove that $f$ is piecewise continuous; more precisely, there is a finite partition of $A$ into $\mathcal{L}^{P}$-definable locally closed subsets $A_{1},\ldots,A_{s}$ of $K^{n}$ such that the restriction of $f$ to each $A_{i}$ is a continuous function.

math.AG↗

arXiv2017

Hölder and Lipschitz continuity of functions definable over Henselian rank one valued fields

Consider a Henselian rank one valued field $K$ of equicharacteristic zero with the three-sorted language $\mathcal{L}$ of Denef--Pas. Let $f: A \to K$ be a continuous $\mathcal{L}$-definable (with parameters) function on a closed bounded subset $A \subset K^{n}$. The main purpose is to prove that then $f$ is Hölder continuous with some exponent $s\geq 0$ and constant $c \geq 0$, a fortiori, $f$ is uniformly continuous. Further, if $f$ is locally Lipschitz continuous with a constant $c$, then $f$ is (globally) Lipschitz continuous with possibly some larger constant $d$. Also stated are some problems concerning continuous and Lipschitz continuous functions definable over Henselian valued fields.

math.AG↗

arXiv2017

A density property of Henselian valued fields

We give an elementary proof of a version of the implicit function theorem over Henselian valued fields $K$. It yields a density property for such fields (introduced in a joint paper with J. Koll{á}r), which is indispensable for ensuring reasonable topological and geometric properties of algebraic subsets of $K^{n}$.

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arXiv2016

Some results from algebraic geometry over complete discretely valued fields

This paper is concerned with algebraic geometry over complete discretely valued fields $K$ of equicharacteristic zero. Several results are given including: the canonical projection $K^{n} \times K\mathbb{P}^{m} \longrightarrow K^{n}$ and blow-ups of the $K$-points of smooth $K$-varieties are definably closed maps, a descent principle, a version of the Lojasiewicz inequality for continuous rational functions, curve selection for semialgebraic sets and the theorem on extending continuous hereditarily rational functions, established for the real field in our joint paper with J. Kollar. Our approach applies the quantifier elimination due to Pas. By the transfer principle of Ax-Kochen-Ershov, all these results carry over to the case of Henselian discretely valued fields. Using different arguments in our subsequent paper, we establish them over Henselian real valued fields of equicharacteristic zero.

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