A Hilbert Lemniscate Theorem in 2  [ Un théorème de la lemniscate de Hilbert dans 2  ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 6, p. 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 : https://doi.org/10.5802/aif.2411
Classification:  32U05,  32W20
Mots 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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {6},
     year = {2008},
     pages = {2191-2220},
     doi = {10.5802/aif.2411},
     mrnumber = {2473634},
     zbl = {1152.32015},
     language = {en},
     url = {http://www.numdam.org/item/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, Tome 58 (2008) no. 6, pp. 2191-2220. doi : 10.5802/aif.2411. http://www.numdam.org/item/AIF_2008__58_6_2191_0/

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