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 $ℳ$.

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...

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.

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.

Nous donnons une preuve géométrique du théorème d’élimination des quantificateurs pour les fonctions logarithmico-exponentielles prouvé initialement par van den Dries, Macintyre et Marker. Notre démonstration n’utilise pas de Théorie des Modèles. Elle repose sur un théorème de préparation pour les fonctions sous-analytiques.

On introduit, dans ce travail, une hypothèse sur le spiralement d’une feuille d’un feuilletage analytique réel de codimension un (hypersurface pfaffienne). On en tire des résultats très généraux de finitude du type de Khovanskii. Des exemples précis montrent la généralité de ces hypersurfaces pfaffiennes. Une description complété des bouts de telles variétés en dimension trois est donnée.

Let $\Omega \subset {ℝ}^n$ be a compact semianalytic set and let $ℱ$ be a collection of real analytic functions defined in some neighbourhood of $\Omega$. Let $Y_{\omega }$ be the germ at $\omega$ of the set ${\bigcap }_{f\in ℱ}{f}^{-1}\left(0\right)$. Then there exist analytic functions $v_1,v_2,...,v_s$ defined in a neighbourhood of $\Omega$ such that $\frac{1}{2}\chi \left(\mathrm{lk}(\omega ,Y_{\omega })\right)={\sum }_{i=1}^s\mathrm{sgn}{v}_i\left(\omega \right)$, for all $\omega \in \Omega$.