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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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The Łojasiewicz exponent of an analytic mapping of two complex variables at an isolated zero

100%
Chadzyński J., Krasiński T.
Banach Center Publications
|
1988
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tom 20
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nr 1
139-146
2
Content available remote

A set on which the Łojasiewicz exponent at infinity is attained

100%
Chądzyński J., Krasiński T.
Annales Polonici Mathematici
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1997
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tom 67
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nr 2
191-197
EN
We show that for a polynomial mapping $F = (f₁,..., fₘ): ℂ^n → ℂ^m$ the Łojasiewicz exponent $𝓛_∞(F)$ of F is attained on the set ${z ∈ ℂ^n: f₁(z) ·...· fₘ(z) = 0}$.
3
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A set on which the local Łojasiewicz exponent is attained

100%
Chądzyński J., Krasiński T.
Annales Polonici Mathematici
|
1997
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tom 67
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nr 3
297-301
EN
Let U be a neighbourhood of 0 ∈ ℂⁿ. We show that for a holomorphic mapping $F = (f₁,..., fₘ): U → ℂ^m$, F(0) = 0, the Łojasiewicz exponent 𝓛₀(F) is attained on the set {z ∈ U: f₁(z)·...·fₘ(z) = 0}.
4
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Intersection of analytic curves

100%
Krasiński T., Nowak K.
Annales Polonici Mathematici
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2003
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tom 80
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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.
5
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Exponent of growth of polynomial mappings of $C^2$ into $C^2$

100%
Chadzyński J., Krasiński T.
Banach Center Publications
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1988
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tom 20
|
nr 1
147-160
6
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The jump of the Milnor number in the X 9 singularity class

100%
Brzostowski S., Krasiński T.
Open Mathematics
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2014
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tom 12
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nr 3
429-435
EN
The jump of the Milnor number of an isolated singularity f 0 is the minimal non-zero difference between the Milnor numbers of f 0 and one of its deformations (f s). We prove that for the singularities in the X 9 singularity class their jumps are equal to 2.
7
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On the Łojasiewicz exponent for analytic curves

100%
Chądzyński J., Krasiński T.
Banach Center Publications
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1998
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tom 44
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nr 1
73-80
EN
An effective formula for the Łojasiewicz exponent for analytic curves in a neighbourhood of 0 ∈ ℂ is given.
8
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On the Łojasiewicz exponent at infinity for polynomial mappings of ℂ² into ℂ² and components of polynomial automorphisms of ℂ²

100%
Chądzyński J., Krasiński T.
Annales Polonici Mathematici
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1992
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tom 57
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nr 3
291-302
EN
A complete characterization of the Łojasiewicz exponent at infinity for polynomial mappings of ℂ² into ℂ² is given. Moreover, a characterization of a component of a polynomial automorphism of ℂ² (in terms of the Łojasiewicz exponent at infinity) is given.
9
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The Łojasiewicz numbers and plane curve singularities

81%
Barroso E., Krasiński T., Płoski A.
Annales Polonici Mathematici
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2005
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tom 87
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nr 1
127-150
EN
For every holomorphic function in two complex variables with an isolated critical point at the origin we consider the Łojasiewicz exponent 𝓛₀(f) defined to be the smallest θ > 0 such that $|grad f(z)| ≥ c|z|^{θ}$ near 0 ∈ ℂ² for some c > 0. We investigate the set of all numbers 𝓛₀(f) where f runs over all holomorphic functions with an isolated critical point at 0 ∈ ℂ².
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