Nous caractérisons, en terme de dimension (topologique et de Hausdorff) des fibres des espaces de limites de tangents et du cône de Whitney, les conditions de régularité $b_{\mathrm{cod}q}$ et $b^{*}$ sur une stratification $C^1$. Nous précisons ces résultats lorsque les espaces qui interviennent ne sont pas fractals, en particulier lorsque la stratification est sous-analytique.

Let $X$ be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of $𝒟$-modules on $X$ induces a fully faithful functor on a subcategory of germs of formal holonomic $𝒟$-modules. Further, given a germ $ℳ$ of holonomic $𝒟$-module, we obtain some results linking the subanalytic sheaf of tempered solutions of $ℳ$ and the classical formal and analytic invariants of $ℳ$.

The Briançon-Skoda number of a ring $R$ is defined as the smallest integer k, such that for any ideal $I\subset R$ and $l\ge 1$, the integral closure of $I^{k+l-1}$ is contained in $I^l$. We compute the Briançon-Skoda number of the local ring of any analytic irreducible planar curve in terms of its Puiseux characteristics. It turns out that this number is closely related to the Milnor number.

Let f:ℝ² → ℝ be a polynomial mapping with a finite number of critical points. We express the degree at infinity of the gradient ∇f in terms of the real branches at infinity of the level curves {f(x,y) = λ} for some λ ∈ ℝ. The formula obtained is a counterpart at infinity of the local formula due to Arnold.

Let $A\subset {ℝ}^n$ be a set-germ at $0\in {ℝ}^n$ such that $0\in \overline{A}$. We say that $r\in S^{n-1}$ is a direction of $A$ at $0\in {ℝ}^n$ if there is a sequence of points $\left\{x_i\right\}\subset A\setminus \left\{0\right\}$ tending to $0\in {ℝ}^n$ such that $\frac{x_i}{\parallel x_i\parallel }\to r$ as $i\to \infty$. Let $D\left(A\right)$ denote the set of all directions of $A$ at $0\in {ℝ}^n$.Let $A,B\subset {ℝ}^n$ be subanalytic set-germs at $0\in {ℝ}^n$ such that $0\in \overline{A}\cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $dim\left(D\right(A)\cap D(B\left)\right)$ is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of $A$ and $B$ are also subanalytic. In particular if two subanalytic...

Si dimostra un risultato di prolungamento per applicazioni meromorfe a valori in uno spazio $q$-completo che generalizza direttamente il risultato classico di Hartogs e migliora risultati di K. Stein.

The aim of this paper is to study the Łojasiewicz exponent of c-holomorphic mappings. After introducing an order of flatness for c-holomorphic mappings we give an estimate of the Łojasiewicz exponent in the case of isolated zero, which is a generalization of the one given by Płoski and earlier by Chądzyński for two variables.

We prove that the infimum of the regular separation exponents of two subanalytic sets at a point is a rational number, and it is also a regular separation exponent of these sets. Moreover, we consider the problem of attainment of this exponent on analytic curves.

We find a relation between the vanishing of a globally defined residue current on ${\mathbb{P}}^n$ and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther’s theorem, for which we also obtain a generalization. Furthermore there are some connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions.

Let X ⊂ (ℝⁿ,0) be a germ of a set at the origin. We suppose X is described by a subalgebra, Cₙ(M), of the algebra of germs of $C^{\infty }$ functions at the origin (see 2.1). This algebra is quasianalytic. We show that the germ X has almost all the properties of germs of semianalytic sets. Moreover, we study the projections of such germs and prove a version of Gabrielov’s theorem.

This is a summary of recent work where we introduced a class of D-modules adapted to study ideals generated by exponential polynomials.