Displaying similar documents to “Multiplicity of polynomials on trajectories of polynomial vector fields in C 3 ”

On blowing up versal discriminants

Piotr Jaworski (1998)

Banach Center Publications

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It is well-known that the versal deformations of nonsimple singularities depend on moduli. The first step in deeper understanding of this phenomenon is to determine the versal discriminant, which roughly speaking is an obstacle for analytic triviality of an unfolding or deformation along the moduli. The goal of this paper is to describe the versal discriminant of Z k , 0 and Q k , 0 singularities basing on the fact that the deformations of these singularities may be obtained as blowing ups of certain...

On the versal discriminant of J k , 0 singularities

Piotr Jaworski (1996)

Annales Polonici Mathematici

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It is well known that the versal deformations of nonsimple singularities depend on moduli. The first step in deeper understanding of this phenomenon is to determine the versal discriminant, which roughly speaking is an obstacle to analytic triviality of an unfolding or deformation along the moduli. The versal discriminant of the Pham singularity ( J 3 , 0 in Arnold’s classification) was thoroughly investigated by J. Damon and A. Galligo [2], [3], [4]. The goal of this paper is to continue their...

On the topological triviality along moduli of deformations of J k , 0 singularities

Piotr Jaworski (2000)

Annales Polonici Mathematici

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It is well known that versal deformations of nonsimple singularities depend on moduli. However they can be topologically trivial along some or all of them. The first step in the investigation of this phenomenon is to determine the versal discriminant (unstable locus), which roughly speaking is the obstacle to analytic triviality. The next one is to construct continuous liftable vector fields smooth far from the versal discriminant and to integrate them. In this paper we extend the results...

On semialgebraic points of definable sets

Artur Piękosz (1998)

Banach Center Publications

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We prove that the semialgebraic, algebraic, and algebraic nonsingular points of a definable set in o-minimal structure with analytic cell decomposition are definable. Moreover, the operation of taking semialgebraic points is idempotent and the degree of complexity of semialgebraic points is bounded.

On the Łojasiewicz exponent for analytic curves

Jacek Chądzyński, Tadeusz Krasiński (1998)

Banach Center Publications

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An effective formula for the Łojasiewicz exponent for analytic curves in a neighbourhood of 0 ∈ ℂ is given.

Intersection theory in complex analytic geometry

Piotr Tworzewski (1995)

Annales Polonici Mathematici

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We present a construction of an intersection product of arbitrary complex analytic cycles based on a pointwise defined intersection multiplicity.

On the Euler characteristic of fibres of real polynomial maps

Adam Parusiński, Zbigniew Szafraniec (1998)

Banach Center Publications

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Let Y be a real algebraic subset of m and F : Y → n be a polynomial map. We show that there exist real polynomial functions g 1 , . . . , g s on n such that the Euler characteristic of fibres of F is the sum of signs of g i .

Decompositions of hypersurface singularities oftype J k , 0

Piotr Jaworski (1994)

Annales Polonici Mathematici

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Applications of singularity theory give rise to many questions concerning deformations of singularities. Unfortunately, satisfactory answers are known only for simple singularities and partially for unimodal ones. The aim of this paper is to give some insight into decompositions of multi-modal singularities with unimodal leading part. We investigate the J k , 0 singularities which have modality k - 1 but the quasihomogeneous part of their normal form only depends on one modulus.