Soit $\mathscr{H}=\bigcup N_i$ la décomposition canonique de l’espace des arcs $\mathscr{H}$ passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que $N_i$ n’est pas inclus dans $N_j$. On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.

We investigate ramification properties with respect to parameters of integrals (distributions) of a class of singular functions over an unbounded cycle which may intersect the singularities of the integrand. We generalize the classical result of Nilsson dealing with the case where the cycle is bounded and contained in the set of holomorphy of the integrand. Such problems arise naturally in the study of exponential representation at infinity of solutions to certain PDE's (see [Z]).

Let U be a neighbourhood of 0 ∈ ℂⁿ. We show that for a holomorphic mapping $F=(f₁,...,fₘ):U\to {ℂ}^m$, F(0) = 0, the Łojasiewicz exponent ₀(F) is attained on the set z ∈ U: f₁(z)·...·fₘ(z) = 0.

Here we show that a Kupka component $K$ of a codimension 1 singular foliation $F$ of $\mathbf{C}{\mathbf{P}}^n,n\ge 6$ with $\mathrm{deg}\left(K\right)$ not a square is a complete intersection. The result implies the existence of a meromorphic first integral of $F$.

Here we show that a Kupka component $K$ of a codimension 1 singular foliation $F$ of $ℂ{\mathbb{P}}^n,n\ge 6$ is a complete intersection. The result implies the existence of a meromorphic first integral of $F$. The result was previously known if $\mathrm{deg}\left(K\right)$ was assumed to be not a square.

To a germ $f:({ℂ}^n,0)\to (ℂ,0)$ with one-dimensional singular locus one associates series of isolated singularities $f_N:=f+l^N$, where l is a general linear function and $N\in$. We prove an attaching result of Iomdin-Lê type which compares the homotopy types of the Milnor fibres of $f_N$ and f. This is a refinement of the Iomdin-Lê theorem in the general setting of a singular underlying space.

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singularity in the projective space of dimension two. Also we calculate higher Melnikov functions under some...

The aim of this note is to give a clearer and more direct proof of the main result of another paper of the author. Moreover, we give some complementary results related to R-complete algebraic foliations with R a rational function of type ℂ*.

Nous commençons par indiquer comment la connaissance du degré d’un opérateur différentiel, unitaire en $s$ et annulant $f^s$, permet de donner un algorithme de calcul du polynôme de Bernstein d’un germe $f$ de fonction analytique à singularité isolée.Nous étudions alors le cas d’une singularité non dégénérée par rapport à son polygôme de Newton; nous donnons un algorithme pour calculer le polynôme de Bernstein de ces singularités et l’équation fonctionnelle associée. Notre méthode utilise une filtration...

Let $(V,0)$ be a germ of a complete intersection variety in ${ℂ}^{n+k}$, $n\>0$, having an isolated singularity at $0$ and $X$ be the germ of a holomorphic vector field having an isolated zero at $0$ and tangent to $V$. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of $X$ is also isolated in the ambient space ${ℂ}^{n+k}$ we give a formula for the homological index in terms of local linear algebra.