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A decomposition of a set definable in an o-minimal structure into perfectly situated sets

Wiesław Pawłucki (2002)

Annales Polonici Mathematici

A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such that any...

A generalization of Radó's theorem

E. M. Chirka (2003)

Annales Polonici Mathematici

If Σ is a compact subset of a domain Ω ⊂ ℂ and the cluster values on ∂Σ of a holomorphic function f in Ω∖Σ, f' ≢ 0, are contained in a compact null-set for the holomorphic Dirichlet class, then f extends holomorphically onto the whole domain Ω.

A linear extension operator for Whitney fields on closed o-minimal sets

Wiesław Pawłucki (2008)

Annales de l’institut Fourier

A continuous linear extension operator, different from Whitney’s, for 𝒞 p -Whitney fields (p finite) on a closed o-minimal subset of n is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.

A note on Bézout's theorem

Sławomir Rams, Piotr Tworzewski, Tadeusz Winiarski (2005)

Annales Polonici Mathematici

We present a version of Bézout's theorem basing on the intersection theory in complex analytic geometry. Some applications for products of surfaces and curves are also given.

A note on Bierstone-Milman-Pawłucki's paper "Composite differentiable functions"

Krzysztof Jan Nowak (2011)

Annales Polonici Mathematici

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.

A note on the Nullstellensatz for c-holomorphic functions

Maciej P. Denkowski (2007)

Annales Polonici Mathematici

We begin this article with a graph theorem and a kind of Nullstellensatz for weakly holomorphic functions. This yields a general Nullstellensatz for c-holomorphic functions on locally irreducible sets. In Section 2 some methods of Płoski-Tworzewski permit us to prove an effective Nullstellensatz for c-holomorphic functions in the case of a proper intersection with the degree of the intersection cycle as exponent. We also extend this result to the case of isolated improper intersection, generalizing...

A proof of the valuation property and preparation theorem

Krzysztof Jan Nowak (2007)

Annales Polonici Mathematici

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 c o n v (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den...

A Schwarz lemma for correspondences and applications.

Kaushal Verma (2003)

Publicacions Matemàtiques

A version of the Schwarz lemma for correspondences is studied. Two applications are obtained namely, the 'non-increasing' property of the Kobayashi metric under correspondences and a weak version of the Wong-Rosay theorem for convex, finite type domains admitting a 'non-compact' family of proper correspondences.

A set on which the Łojasiewicz exponent at infinity is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

We show that for a polynomial mapping F = ( f , . . . , f ) : n m the Łojasiewicz exponent ( F ) of F is attained on the set z n : f ( z ) · . . . · f ( z ) = 0 .

Algèbres analytiques topologiquement noéthériennes. Théorie de Khovanskii

Jean-Claude Tougeron (1991)

Annales de l'institut Fourier

On étudie certaines algèbres de fonctions analytiques réelles définies sur un ouvert Ω de R n . La propriété principale de ces algèbres est que tout semi-analytique de Ω défini globalement à l’aide d’un nombre fini de fonctions de 𝒪 ( Ω ) , admet un nombre fini de composantes connexes. En reprenant les idées de Khovanskii (lemme de Rolle généralisé), on démontre que ces algèbres restent topologiquement noethériennes quand on leur adjoint les solutions de certaines équations différentielles du ler ordre. Par...

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