Les amibesdes variétés algébriques dans ${\left({ℂ}^{*}\right)}^n$ sont les images de ces variétés par l’application des moments $\mathrm{Log}:{\left({ℂ}^{*}\right)}^n\to {ℝ}^n$, $\mathrm{Log}:(z_1,...,z_n)\mapsto (log|z_1|,...,log|z_n\left|\right)$. Des résultats obtenus par G. Mikhalkin montrent l’utilité des amibes pour l’étude des variétés algébriques réelles et complexes. Les amibes peuvent être déformées en des complexes polyédraux appelésvariétés algébriques tropicales. Cette déformation permet, en particulier, de calculer les invariants de Gromov-Witten du plan projectif et d’autres surfaces toriques en dénombrant des courbes...

Completeness of a dilation system ${\left(\varphi \left(nx\right)\right)}_{n\ge 1}$ on the standard Lebesgue space $L^2(0,1)$ is considered for 2-periodic functions $\varphi$. We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space $H^2\left({𝔻}_2^{\infty }\right)$ on the Hilbert multidisc ${𝔻}_2^{\infty }$. Several simple sufficient conditions are exhibited, which include however practically all previously known results (Wintner; Kozlov; Neuwirth, Ginsberg, and Newman; Hedenmalm, Lindquist, and Seip). For instance, each of the following conditions implies cyclicity...

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...

The local Nullstellensatz exponent for holomorphic mappings via intersection theory for the cases of isolated and quasi-complete intersection is considered.

Sufficient conditions for the existence of an analytic solution of analytic equations in the complex and real cases are given.

Our objective is to construct residue currents from Bochner-Martinelli type kernels; the computations hold in the non complete intersection case and provide a new and more direct approach of the residue of Coleff-Herrera in the complete intersection case; computations involve crucial relations with toroidal varieties and multivariate integrals of the Mellin-Barnes type.

We show that for an interpolating sequence in the polydisk one can construct a universal divisor for Hardy spaces.