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A note on singularities at infinity of complex polynomials

Adam Parusiński (1997)

Banach Center Publications

Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family $\bar{f}$ of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the $C^{\infty}$ -triviality of f. If the support of sheaf of vanishing cycles of $\bar{f}$ is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in...

Analytic equivalence of plane curve singularities $y^{n} + x^{α} y + x^{β} a (x) = 0$ .

Stepanović, B., Lipkovski, A. (2007)

Publications de l'Institut Mathématique. Nouvelle Série

Bi-Lipschitz trivialization of the distance function to a stratum of a stratification

Adam Parusiński (2005)

Annales Polonici Mathematici

Given a Lipschitz stratification 𝒳 that additionally satisfies condition (δ) of Bekka-Trotman (for instance any Lipschitz stratification of a subanalytic set), we show that for every stratum N of 𝒳 the distance function to N is locally bi-Lipschitz trivial along N. The trivialization is obtained by integration of a Lipschitz vector field.

Characterization of non-degenerate plane curve singularities.

García Barroso, Evelia R., Lenarcik, Andrzej, Płoski, Arkadiusz (2007)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

Cône normal et régularités de Kuo-Verdier

Patrice Orro, David Trotman (2002)

Bulletin de la Société Mathématique de France

Nous introduisons de nouvelles régularités de Kuo-Verdier $(r^{e})$ et montrons que pour une stratification $C^{2}$ $(a + r^{e}) ...$

Ein Torelli-Satz für die unimodalen und bimodularen Hyperflächensingularitäten.

Claus Hertling (1995) Mathematische Annalen

Equianalytic and equisingular families of curves on surfaces.

Gert-Martin Greuel, Christoph Lossen (1996) Manuscripta mathematica

Équisingularité réelle II : invariants locaux et conditions de régularité

Georges Comte, Michel Merle (2008) Annales scientifiques de l'École Normale Supérieure

On définit, pour un germe d’ensemble sous-analytique, deux nouvelles suites finies d’invariants numériques. La première a pour termes les localisations des courbures de Lipschitz-Killing classiques, la seconde est l’équivalent réel des caractéristiques évanescentes complexes introduites par M. Kashiwara. On montre que chaque terme d’une de ces suites est combinaison linéaire des termes de l’autre, puis on relie ces invariants à la géométrie des discriminants des projections du germe sur des plans...

Équisingularité réelle : nombres de Lelong et images polaires

Georges Comte (2000) Annales scientifiques de l'École Normale Supérieure

Fibrations associées à un pinceau de courbes planes

Hélène Maugendre, Françoise Michel (2001) Annales de la Faculté des sciences de Toulouse : Mathématiques

Fronces et doubles plis

Jean-Pierre Henry, Michel Merle (1996) Compositio Mathematica

Invariants of bi-Lipschitz equivalence of real analytic functions

Jean-Pierre Henry, Adam Parusiński (2004) Banach Center Publications

We construct an invariant of the bi-Lipschitz equivalence of analytic function germs (ℝⁿ,0) → (ℝ,0) that varies continuously in many analytic families. This shows that the bi-Lipschitz equivalence of analytic function germs admits continuous moduli. For a germ f the invariant is given in terms of the leading coefficients of the asymptotic expansions of f along the sets where the size of |x| |grad f(x)| is comparable to the size of |f(x)|.

Modular deformations and space curve singularities.

Bernd Martin (2003) Revista Matemática Iberoamericana

We investigate different concepts of modular deformations of germs of isolated singularities (infinitesimal, Artinian, formal). An obstruction calculus based on the graded Lie algebra structure of the tangent cohomology for modular dcformations is introduced. As the main result the characterisation of the maximal infinitesimally modular subgerm of the miniversal family as flattening stratum of the relative Tjurina module is extended from ICIS to space curve singularities.

Modules de feuillages holomorphes singuliers: I équisingularité.

J.F. Mattei (1991) Inventiones mathematicae

Motivic-type invariants of blow-analytic equivalence

Satoshi Koike, Adam Parusiński (2003) Annales de l'Institut Fourier

To a given analytic function germ

f : (ℝ^{d}, 0) \to (ℝ, 0)

, we associate zeta functions

Z_{f, +}

Z_{f, -} \in ℤ [[T]]

, defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic...

Multiplicieties and equisingularity of ICIS germs.

Terence Gaffney (1996)

Inventiones mathematicae

On asymptotic critical values and the Rabier Theorem

Zbigniew Jelonek (2004)

Banach Center Publications

Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f ₁, . . ., f_{m}) : X \to k^{m}$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function...

On blowing up versal discriminants

Piotr Jaworski (1998)

Banach Center Publications

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 deformations...

On the topological triviality along moduli of deformations of $J_{k, 0}$ singularities

Piotr Jaworski (2000)

Annales Polonici Mathematici

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 of J....

On the versal discriminant of $J_{k, 0}$ singularities

Piotr Jaworski (1996)

Annales Polonici Mathematici

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 work and...

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