Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure ${\mu }_S:={\sum }_{a\in S}(1-|a\left|²\right)ⁿ{\delta }_a$ is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure ${\mu }_S$ bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual bounded sequence...

A sufficient condition is given to make a sequence of hyperplanes in the complex unit ball an interpolating sequence for $H^{\infty }$, i.e. bounded holomorphic functions on the hyperplanes can be boundedly extended.

Let $K$ be a compact subset of an hyperconvex open set $D\subset {ℂ}^n$, forming with D a Runge pair and such that the extremal p.s.h. function ω(·,K,D) is continuous. Let H(D) and H(K) be the spaces of holomorphic functions respectively on D and K equipped with their usual topologies. The main result of this paper contains as a particular case the following statement: if T is a continuous linear map of H(K) into H(K) whose restriction to H(D) is continuous into H(D), then the restriction of T to $H\left(D_{\alpha }\right)$ is a continuous...

On démontre un résultat concernant l’interpolation de fonctions analytiques sur une perturbation d’ensemble produit qui, dans le cas $p$-adique, répond à une conjecture de P.Robba et, dans le cas complexe, complète des résultats antérieurs de E.Bombieri, S.Lang, D.Masser, J.-C.Moreau et M.Waldschmidt.

We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain $D$ of dimension $N$. If $\Sigma \subset bD$ is a smooth manifold of dimension $N$ and a maximum modulus set, then it admits a unique foliation by compact interpolation manifolds. There is a semiglobal converse in the real analytic case. Two functions in $A^2\left(D\right)$ with the same smooth $N$-dimensional maximum modulus set are analytically related and are polynomially related if a certain homology class in $H_1(D,\mathbf{R})$ vanishes or if $\bar{D}\subset {\mathbf{C}}^N$ is polynomially...

Soit $K$ un compact polynomialement convexe de ${\mathbf{C}}^n$ et $V_K$ son “potentiel logarithmique extrémal” dans ${\mathbf{C}}^n$. Supposons que $K$ est régulier (i.e. $V_K$ continue) et soit $f$ une fonction holomorphe sur un voisinage de $K$. On construit alors une suite $\{{\mathbf{P}}_{\ell }{\}}_{\ell \ge 1}$ de polynôme de $n$ variables complexes avec deg$({\mathbf{P}}_{)}\le \ell$ pour $\ell \ge 1$, telle que l’erreur d’approximation ${\max }_{z\in K}|f\left(z\right)-P_{\ell }\left(z\right)|$ soit contrôlée de façon assez précise en fonction du “pseudorayon de convergence” de $f$ par rapport à $K$ et du degré de convergence $\ell$. Ce résultat est ensuite utilisé pour étendre...

We give a state-of-the-art survey of investigations concerning multivariate polynomial inequalities. A satisfactory theory of such inequalities has been developed due to applications of both the Gabrielov-Hironaka-Łojasiewicz subanalytic geometry and pluripotential methods based on the complex Monge-Ampère operator. Such an approach permits one to study various inequalities for polynomials restricted not only to nice (nonpluripolar) compact subsets of ℝⁿ or ℂⁿ but also their versions for pieces...

Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989.In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations...

We show that the converse of the aproximation theorem of Andreotti and Grauert does not hold. More precisely we construct a $4$-complete open subset $D\subset {ℂ}^6$ (which is an analytic complement in the unit ball) such that the restriction map $H^3({ℂ}^6,ℱ)\to H^3(D,ℱ)$ has a dense image for every $ℱ\in Coh\left({ℂ}^6\right)$ but the pair $(D,{ℂ}^6)$ is not a $4$-Runge pair.

By the Oka-Weil theorem, each holomorphic function f in a neighbourhood of a compact polynomially convex set $K\subset {ℂ}^N$ can be approximated uniformly on K by complex polynomials. The famous Bernstein-Walsh-Siciak theorem specifies the Oka-Weil result: it states that the distance (in the supremum norm on K) of f to the space of complex polynomials of degree at most n tends to zero not slower than the sequence M(f)ρ(f)ⁿ for some M(f) > 0 and ρ(f) ∈ (0,1). The aim of this note is to deduce the uniform version,...

We prove several new results on the multivariate transfinite diameter and its connection with pluripotential theory: a formula for the transfinite diameter of a general product set, a comparison theorem and a new expression involving Robin's functions. We also study the transfinite diameter of the pre-image under certain proper polynomial mappings.