arXiv subjects

Krzysztof Jan Nowak

Publications and source records attributed to Krzysztof Jan Nowak.

25 records · Page 2Linked to original sources

arXiv2016

Some results from algebraic geometry over Henselian real valued fields

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 shrinking (a relaxed version of curve selection) for definable sets, the canonical projection $K^{n} \times K\mathbb{P}^{m} \to K^{n}$ and blow-ups of the $K$-points of smooth $K$-varieties are definably closed maps, a descent property for blow-ups, a version of the Lojasiewicz inequality for continuous rational functions and the theorem on extending continuous hereditarily rational functions, established for the real and $p$-adic varieties in our joint paper with J. Kollar. The descent property enables application of desingularization and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach applies quantifier elimination due to Pas.

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arXiv2016

Some results of algebraic geometry over Henselian rank one valued fields

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 blow-ups of the $K$-rational points of smooth $K$-varieties are definably closed maps, a descent property for blow-ups, curve selection for definable sets, a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the $K$-topology and extending continuous hereditarily rational functions, established for the real and $p$-adic varieties in our joint paper with J. Kollár. The descent property enables application of resolution of singularities and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach relies on quantifier elimination due to Pas and a concept of fiber shrinking for definable sets, which is a relaxed version of curve selection. The last three sections are devoted to the theory of regulous functions and sets over such valued fields. Regulous geometry over the real ground field $\mathbb{R}$ was developed by Fichou--Huisman--Mangolte--Monnier. The main results here are regulous versions of Nullstellensatz and Cartan's Theorems A and B.

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arXiv2015

Quantifier elimination in quasianalytic structures via non-standard analysis

The paper is a continuation of our earlier article where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented here. We also provide a quasianalytic exposition of the results concerning rectilinearization of terms and of definable functions from our earlier research. It will be used to demonstrate a quasianalytic structure corresponding to a Denjoy-Carleman class which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented merely by the reciprocal function. More precisely, we construct a plane definable curve, which indicates both that the classical theorem by J. Denef and L. van den Dries as well as Łojasiewicz's theorem that every subanalytic curve is semianalytic are no longer true for quasianalytic structures. Besides rectilinearization of terms, our construction makes use of some theorems on power substitution for Denjoy-Carleman classes and on non-extendability of quasianalytic function germs. The last result relies on Grothendieck's factorization and open mapping theorems for (LF)-spaces. Note finally that this paper comprises our earlier preprints on the subject from May 2012.

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arXiv2014

A counterexample concerning quantifier elimination in quasianalytic structures

This paper is a revised version of our preprints IMUJ Preprint 2012/04 and RAAG Preprint 343 from May 2012. It provides an example of a quasianalytic structure which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented by the reciprocal function. It also demonstrates that Lojasiewicz's theorem that every subanalytic curve is semianalytic is no longer true in quasianalytic structures. Our construction applies rectilinearization of terms, established in our earlier papers, as well as some theorems on power substitution for Denjoy-Carleman classes and on non-extendability of quasianalytic function germs. The last result relies on Grothendieck's factorization and open mapping theorems for (LF)-spaces.

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arXiv2014

Generic intersections of differentiable submanifolds

Consider a transitive action of a Lie group $G$ on a (real analytic) manifold $M$ of dimension $m$, and two (embedded) submanifolds $A$ and $B$ in $M$ of sufficiently large class and of dimension $k$ and $l$, respectively. We prove that, for a generic $σ\in G$, the intersection $σ(A) \cap B$ is transversal, whence a submanifold of dimension $k+l-m$ or the empty set. This paper is a continuation of our previous article, devoted to definable transitive actions of definable groups on manifolds and generic intersections in an o-minimal structure, and inspired by a question of Jan Mycielski about the intersections of translates of analytic sets in the real plane.

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arXiv2013

On division of quasianalytic function germs

In this paper, we establish the following criterion for divisibility in the local ring of those quasianalytic function germs at zero which are definable in a polynomially bounded structure. A sufficient (and necessary) condition for the divisibility of two such function germs is that of their Taylor series at zero in the formal power series ring.

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arXiv2012

On hereditarily rational functions

In this paper, we give a short proof of a theorem by Kollár on hereditarily rational functions. This is an answer to his appeal to find an elementary proof which does not rely so much on resolution of singularities. Our approach does not make use of desingularization techniques. Instead, we apply a stronger version of the Łojasiewicz inequality. Moreover, this allows us to sharpen Kollár's theorem.

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