We prove that any divisor $Y$ of a global analytic set $X\subset {ℝ}^n$ has a generic equation, that is, there is an analytic function vanishing on $Y$ with multiplicity one along each irreducible component of $Y$. We also prove that there are functions with arbitrary multiplicities along $Y$. The main result states that if $X$ is pure dimensional, $Y$ is locally principal, $X/Y$ is not connected and $Y$ represents the zero class in $H_{q-1}^{\infty }(X,{\mathbb{Z}}_2)$ then the divisor $Y$ is globally principal.

We present here three examples concerning polynomial hulls of some manifolds in C2.1. Some real surfaces with equation w = P (z,z') + G(z) where P is a homogeneous polynomial of degree n and G(z) = o(|z|n) at 0 which are locally polynomially convex at 0.2. Some real surfaces MF with equation w = zn+kz'n + F(z,z') such that the hull of Mf ∩ B'(0,1) contains a neighbourhood of 0.3. A contable union of totally real planes (Pj) such that B'(0,1) ∩ (∪j∈N Pj) is polynomially convex.

We extend a result of M. Tamm as follows:Let $f:A\to ℝ,A\subseteq {ℝ}^{m+n}$, be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x\mapsto x^r:(0,\infty )\to ℝ,r\in ℝ$. Then there exists $N\in \mathbb{N}$ such that for all $(a,b)\in A$, if $y\mapsto f(a,y)$ is $C^N$ in a neighborhood of $b$, then $y\mapsto f(a,y)$ is real analytic in a neighborhood of $b$.

Nous définissons deux notions nouvelles en géométrie analytique réelle, celle de fonction Nash-analytique et celle de faisceau semi-cohérent. Avec ces notions, nous obtenons des théorèmes de cohérence analogues à ceux du cas complexe (théorème de cohérence d’Oka, théorème de l’image directe, cohérence d’un ensemble analytique complexe).

The Gleason problem is solved on real analytic pseudoconvex domains in ${\mathbf{C}}^2$. In this case the weakly pseudoconvex points can be a two-dimensional subset of the boundary. To reduce the Gleason problem to a $\bar{\partial }$ question it is shown that the set of Kohn-Nirenberg points is at most one-dimensional. In fact, except for a one-dimensional subset, the weakly pseudoconvex boundary points are $R$-points as studied by Range and therefore allow local sup-norm estimates for $\bar{\partial }$.

We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$. Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$, (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_1,\cdots ,u_m):U\to {ℝ}^m$ such that $U\cap X=\{u_1\cdots u_r=0\}$, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash manifold...

Soit $\pi :V\to W$ un morphisme propre fini et surjectif entre deux variétés analytiques complexes. Nous donnons une caractérisation des fonctions (continues) sur $W$ qui sont de la forme ${\pi }_{*}f$ où $f$ est une fonction $C^{\infty }$ sur $V$. Pour cela nous introduisons la notion de fonction de type trace sur une variété analytique complexe. Ces fonctions sont analytiques réelles en dehors d’une hypersurface complexe et admettent des singularités très simples aux points de cette hypersurface.