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1
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Remarks on the generalized index of an analytic improper intersection

100%
Nowak K.
Annales Polonici Mathematici
|
2003
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tom 81
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nr 1
47-53
EN
This article continues the investigation of the analytic intersection algorithm from the perspective of deformation to the normal cone, carried out in the previous papers of the author [7, 8, 9]. The main theorem asserts that, given an analytic set V and a linear subspace S, every collection of hyperplanes, admissible with respect to an algebraic bicone B, realizes the generalized intersection index of V and S. This result is important because the conditions for a collection of hyperplanes to be admissible with respect to B are of geometric nature: it is not necessary to analyse the embedded components of the intersections involved, but only the supports of the intersections of B with successive hyperplanes.
2
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Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices

100%
Nowak K.
Annales Polonici Mathematici
|
2011
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tom 101
|
nr 3
275-291
EN
This paper investigates hyperbolic polynomials with quasianalytic coefficients. Our main purpose is to prove factorization theorems for such polynomials, and next to generalize the results of K. Kurdyka and L. Paunescu about perturbation of analytic families of symmetric matrices to the quasianalytic setting.
3
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On arc-analytic functions definable by a Weierstrass system

100%
Nowak K.
Annales Polonici Mathematici
|
2011
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tom 100
|
nr 1
99-104
EN
This paper presents certain characterizations through blowing up of arc-analytic functions definable by a convergent Weierstrass system closed under complexification.
4
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Supplement to the paper "Improper intersections in complex analytic geometry" (Dissertationes Math. 391 (2001))

100%
Nowak K.
Annales Polonici Mathematici
|
2001
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tom 76
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nr 3
5
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The Abhyankar-Jung theorem for excellent henselian subrings of formal power series

100%
Nowak K.
Annales Polonici Mathematici
|
2010
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tom 98
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nr 3
221-229
EN
Given an algebraically closed field K of characteristic zero, we prove the Abhyankar-Jung theorem for any excellent henselian ring whose completion is a formal power series ring K[[z]]. In particular, examples include the local rings which form a Weierstrass system over the field K.
6
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Addendum to the paper "Decomposition into special cubes and its application to quasi-subanalytic geometry" (Ann. Polon. Math. 96 (2009), 65-74)

100%
Nowak K.
Annales Polonici Mathematici
|
2010
|
tom 98
|
nr 2
201-205
7
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Division of Distributions by Locally Definable Quasianalytic Functions

100%
Nowak K.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2010
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tom 58
|
nr 3
201-208
EN
We demonstrate that the Łojasiewicz theorem on the division of distributions by analytic functions carries over to the case of division by quasianalytic functions locally definable in an arbitrary polynomially bounded, o-minimal structure which admits smooth cell decomposition. Hence, in particular, the principal ideal generated by a locally definable quasianalytic function is closed in the Fréchet space of smooth functions.
8
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On a universal axiomatization of the real closed fields

100%
Nowak K.
Annales Polonici Mathematici
|
1996-1997
|
tom 65
|
nr 1
95-103
EN
This paper presents a natural axiomatization of the real closed fields. It is universal and admits quantifier elimination.
9
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A note on Bierstone-Milman-Pawłucki's paper "Composite differentiable functions"

100%
Nowak K.
Annales Polonici Mathematici
|
2011
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tom 102
|
nr 3
293-299
EN
We demonstrate that the composite function theorems of Bierstone-Milman-Pawłucki and of Glaeser carry over to any polynomially bounded, o-minimal structure which admits smooth cell decomposition. Moreover, the assumptions of the o-minimal versions can be considerably relaxed compared with the classical analytic ones.
10
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Rectilinearization of functions definable by a Weierstrass system and its applications

100%
Nowak K.
Annales Polonici Mathematici
|
2010
|
tom 99
|
nr 2
129-141
EN
This paper presents several theorems on the rectilinearization of functions definable by a convergent Weierstrass system, as well as their applications to decomposition into special cubes and quantifier elimination.
11
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Supplement to the paper "Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices" (Ann. Polon. Math. 101 (2011), 275-291)

100%
Nowak K.
Annales Polonici Mathematici
|
2012
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tom 103
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nr 1
101-107
EN
In IMUJ Preprint 2009/05 we investigated the quasianalytic perturbation of hyperbolic polynomials and symmetric matrices by applying our quasianalytic version of the Abhyankar-Jung theorem from IMUJ Preprint 2009/02, whose proof relied on a theorem by Luengo on ν-quasiordinary polynomials. But those papers of ours were suspended after we had become aware that Luengo's paper contained an essential gap. This gave rise to our subsequent article on quasianalytic perturbation theory, which developed, however, different methods and techniques. A recent paper by Parusiński-Rond validates Luengo's result, which allows us to resume our previous approach.
12
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A proof of the valuation property and preparation theorem

100%
Nowak K.
Annales Polonici Mathematici
|
2007
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tom 92
|
nr 1
75-85
EN
The purpose of this article is to present a short model-theoretic proof of the valuation property for a polynomially bounded o-minimal theory T. The valuation property was conjectured by van den Dries, and proved for the polynomially bounded case by van den Dries-Speissegger and for the power bounded case by Tyne. Our proof uses the transfer principle for the theory $T_{conv}$ (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den Dries-Lewenberg. The main tools applied here are saturation, the Marker-Steinhorn theorem on parameter reduction and heir-coheir amalgams. The significance of the valuation property lies to a great extent in its geometric content: it is equivalent to the preparation theorem which says, roughly speaking, that every definable function of several variables depends piecewise on any fixed variable in a certain simple fashion. The latter originates in the work of Parusiński for subanalytic functions, and of Lion-Rolin for logarithmic-exponential functions. Van den Dries-Speissegger have proved the preparation theorem in the o-minimal setting (for functions definable in a polynomially bounded structure or logarithmic-exponential over such a structure). Also, the valuation property makes it possible to establish quantifier elimination for polynomially bounded expansions of the real field ℝ with exponential function and logarithm.
13
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On the Euler characteristic of the links of a set determined by smooth definable functions

100%
Nowak K.
Annales Polonici Mathematici
|
2008
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tom 93
|
nr 3
231-246
EN
The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ($C^{∞}$) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := {u ∈ U: φ(u) ≤ 0}, Z := {u ∈ U: φ(u) = 0}. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven that the Euler characteristic of those links (being a local topological invariant) can be expressed as a finite sum of the signs of global smooth definable functions: $χ(lk(u;W)) = ∑_{i=1}^{r} sgn σ_{i}(u)$, $1/2χ(lk(u;Z)) = ∑_{i=1}^{s} sgnζ_{i}(u)$. We also present a version for functions depending smoothly on a parameter. The analytic case of these formulae has been worked out by Nowel. As an immediate consequence, the Euler characteristic of each link of the zero set Z is even. This generalizes to the o-minimal setting a classical result of Sullivan about real algebraic sets.
14
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Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings

100%
Nowak K.
Annales Polonici Mathematici
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2009
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tom 96
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nr 3
247-282
EN
This paper investigates the geometry of the expansion $𝓡_{Q}$ of the real field ℝ by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński-Lion-Rolin). To this end, we study non-standard models 𝓡 of the universal diagram T of $𝓡_{Q}$ in the language ℒ augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which the majority of fundamental results in analytic geometry rely, but which is unavailable in the general quasianalytic geometry. The basic tools applied here are transformation to normal crossings and decomposition into special cubes. The latter method, developed in our earlier article [Ann. Polon. Math. 96 (2009), 65-74], combines modifications by blowing up with a suitable partitioning. Via an analysis of ℒ-terms and infinitesimals, we prove the valuation property for functions given by ℒ-terms, and next the exchange property for substructures of a given model 𝓡. Our proofs are based on the concepts of analytically independent as well as active and non-active infinitesimals, introduced in this article. Further, quantifier elimination for T is established through model-theoretic compactness. The universal theory T is thus complete and o-minimal, and $𝓡_{Q}$ is its prime model. Under the circumstances, every definable function is piecewise given by ℒ-terms, and therefore the previous results concerning ℒ-terms generalize immediately to definable functions. In this fashion, we obtain the valuation property and preparation theorem for quasi-subanalytic functions. Finally, a quasi-subanalytic version of Puiseux's theorem with parameter is demonstrated.
15
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Decomposition into special cubes and its applications to quasi-subanalytic geometry

100%
Nowak K.
Annales Polonici Mathematici
|
2009
|
tom 96
|
nr 1
65-74
EN
The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.
16
Content available remote

A theorem on generic intersections in an o-minimal structure

100%
Nowak K.
Fundamenta Mathematicae
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2014
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tom 227
|
nr 1
21-25
EN
Consider a transitive definable action of a Lie group G on a definable manifold M. Given two (locally) definable subsets A and B of M, we prove that the dimension of the intersection σ(A) ∩ B is not greater than the expected one for a generic σ ∈ G.
17
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Quantifier elimination in quasianalytic structures via non-standard analysis

100%
Nowak K.
Annales Polonici Mathematici
|
2015
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tom 114
|
nr 3
235-267
EN
The paper is a continuation of an earlier one 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 quasianalytic 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 1/x. More precisely, we construct a definable plane curve, which indicates that both 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.
18
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Improper intersections in complex analytic geometry

80%
Nowak K.
Instytut Matematyczny Polskiej Akademii Nauk
19
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Intersection of analytic curves

64%
Krasiński T., Nowak K.
Annales Polonici Mathematici
|
2003
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tom 80
|
nr 1
193-202
EN
We give a relation between two theories of improper intersections, of Tworzewski and of Stückrad-Vogel, for the case of algebraic curves. Given two arbitrary quasiprojective curves V₁,V₂, the intersection cycle V₁ ∙ V₂ in the sense of Tworzewski turns out to be the rational part of the Vogel cycle v(V₁,V₂). We also give short proofs of two known effective formulae for the intersection cycle V₁ ∙ V₂ in terms of local parametrizations of the curves.
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