A Hilbert Lemniscate Theorem in 2
Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 2191-2220.

For a regular, compact, polynomially convex circled set K in C 2 , we construct a sequence of pairs {P n ,Q n } of homogeneous polynomials in two variables with deg P n = deg Q n =n such that the sets K n :={(z,w)C 2 :|P n (z,w)|1,|Q n (z,w)|1} approximate K and if K is the closure of a strictly pseudoconvex domain the normalized counting measures associated to the finite set {P n =Q n =1} converge to the pluripotential-theoretic Monge-Ampère measure for K. The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex variable.

Pour un compact K dans C 2 , regulier, pôlynomiallement convexe et cerclé, on construit une suite de paires {P n ,Q n } avec P n ,Q n pôlynomes homogènes en deux variables et deg P n = deg Q n =n tel que les ensembles K n :={(z,w)C 2 :|P n (z,w)|1,|Q n (z,w)|1} font une approximation de K et quand K est la fermeture d’un domaine strictement pseudoconvexe les mesures de comptage normalisées associées à l’ensemble fini {P n =Q n =1} tendent vers la mesure de Monge-Ampère pour K. L’élément principal est un théorème d’approximation pour les fonctions sousharmoniques de croissance logarithmique à une variable.

DOI: 10.5802/aif.2411
Classification: 32U05,  32W20
Keywords: Logarithmic potential, Monge-Ampère measure, subharmonic functions, atomization
Bloom, Thomas 1; Levenberg, Norman 2; Lyubarskii, Yu. 3

1 University of Toronto Toronto (Canada)
2 Indiana University Bloomington, IN 47405 (USA)
3 Norwegian University of Science and Technology Trondheim, 7491 (Norway)
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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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