no. 6
A Hilbert Lemniscate Theorem in ℂ 2
[Un théorème de la lemniscate de Hilbert dans ℂ 2 ]
Bloom, Thomas1 ; Levenberg, Norman2 ; 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)
Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2191-2220.

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.

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.

DOI : 10.5802/aif.2411
Classification : 32U05, 32W20
Keywords: Logarithmic potential, Monge-Ampère measure, subharmonic functions, atomization
Mot clés : potentiel logarithmique, mesure de Monge-Ampère, fonctions sousharmoniques, atomisation

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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     title = {A {Hilbert} {Lemniscate} {Theorem} in $\mathbb{C}^2$},
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Bloom, Thomas; Levenberg, Norman; Lyubarskii, Yu. A Hilbert Lemniscate Theorem in $\mathbb{C}^2$. Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2191-2220. doi : 10.5802/aif.2411. https://aif.centre-mersenne.org/articles/10.5802/aif.2411/

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