A note on singularities at infinity of complex polynomials

Adam Parusiński

Displaying similar documents to “A note on singularities at infinity of complex polynomials”

Irregular fibers of complex polynomials in two variables.

Arnaud Bodin (2004)

Revista Matemática Complutense

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For a complex polynomial in two variables we study the morphism induced in homology by the embedding of an irregular fiber in a regular neighborhood of it. We give necessary and sufficient conditions for this morphism to be injective, surjective. Particularly this morphism is an isomorphism if and only if the corresponding irregular value is regular at infinity. We apply these results to the study of vanishing and invariant cycles.

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...

The Milnor number of functions on singular hypersurfaces

Mariusz Zając (1996)

Banach Center Publications

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The behaviour of a holomorphic map germ at a critical point has always been an important part of the singularity theory. It is generally known (cf. [5]) that we can associate an integer invariant - called the multiplicity - to each isolated critical point of a holomorphic function of many variables. Several years later it was noticed that similar invariants exist for function germs defined on isolated hypersurface singularities (see [1]). The present paper aims to show a simple approach...

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.

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...

Singularities of the maximum function over a preimage

Aleksey Davydov (1995)

Banach Center Publications

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Deformation of polar methods

David B. Massey, Dirk Siersma (1992)

Annales de l'institut Fourier

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We study deformations of hypersurfaces with one-dimensional singular loci by two different methods. The first method is by using the Le numbers of a hypersurfaces singularity — this falls under the general heading of a “polar” method. The second method is by studying the number of certain special types of singularities which occur in generic deformations of the original hypersurface. We compare and contrast these two methods, and provide a large number of examples.

Semi-simple Carrousels and the Monodromy

David B. Massey (2006)

Annales de l’institut Fourier

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Let $𝒰$ be an open neighborhood of the origin in $ℂ^{n + 1}$ and let $f : (𝒰, 0) \to (ℂ, 0)$ be complex analytic. Let $z_{0}$ be a generic linear form on $ℂ^{n + 1}$ . If the relative polar curve $Γ_{f, z_{0}}^{1}$ at the origin is irreducible and the intersection number $(Γ_{f, z_{0}}^{1} {\cdot V (f))}_{0}$ is prime, then there are severe restrictions on the possible degree $n$ cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when $(Γ_{f, z_{0}}^{1} {\cdot V (f))}_{0}$ is not prime.

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...