For a regular, compact, polynomially convex circled set in , we construct a sequence of pairs of homogeneous polynomials in two variables with such that the sets approximate and if is the closure of a strictly pseudoconvex domain the normalized counting measures associated to the finite set converge to the pluripotential-theoretic Monge-Ampère measure for . The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex variable.
Pour un compact dans , regulier, pôlynomiallement convexe et cerclé, on construit une suite de paires avec pôlynomes homogènes en deux variables et tel que les ensembles font une approximation de et quand est la fermeture d’un domaine strictement pseudoconvexe les mesures de comptage normalisées associées à l’ensemble fini tendent vers la mesure de Monge-Ampère pour . L’élément principal est un théorème d’approximation pour les fonctions sousharmoniques de croissance logarithmique à une variable.
Keywords: Logarithmic potential, Monge-Ampère measure, subharmonic functions, atomization
Mot clés : potentiel logarithmique, mesure de Monge-Ampère, fonctions sousharmoniques, atomisation
@article{AIF_2008__58_6_2191_0, author = {Bloom, Thomas and Levenberg, Norman and Lyubarskii, Yu.}, title = {A {Hilbert} {Lemniscate} {Theorem} in $\mathbb{C}^2$}, journal = {Annales de l'Institut Fourier}, pages = {2191--2220}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {6}, year = {2008}, doi = {10.5802/aif.2411}, zbl = {1152.32015}, mrnumber = {2473634}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2411/} }
TY - JOUR AU - Bloom, Thomas AU - Levenberg, Norman AU - Lyubarskii, Yu. TI - A Hilbert Lemniscate Theorem in $\mathbb{C}^2$ JO - Annales de l'Institut Fourier PY - 2008 SP - 2191 EP - 2220 VL - 58 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2411/ DO - 10.5802/aif.2411 LA - en ID - AIF_2008__58_6_2191_0 ER -
%0 Journal Article %A Bloom, Thomas %A Levenberg, Norman %A Lyubarskii, Yu. %T A Hilbert Lemniscate Theorem in $\mathbb{C}^2$ %J Annales de l'Institut Fourier %D 2008 %P 2191-2220 %V 58 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2411/ %R 10.5802/aif.2411 %G en %F AIF_2008__58_6_2191_0
Bloom, Thomas; Levenberg, Norman; Lyubarskii, Yu. A Hilbert Lemniscate Theorem in $\mathbb{C}^2$. Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 2191-2220. doi : 10.5802/aif.2411. http://archive.numdam.org/articles/10.5802/aif.2411/
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