A Hilbert Lemniscate Theorem in ${ℂ}^2$

Pour un compact $K$ dans ${\mathbf{C}}^2$, 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 $\mathrm{deg}{P}_n=$ $\mathrm{deg}{Q}_n=n$ tel que les ensembles $K_n:=\{(z,w)\in {\mathbf{C}}^2:|P_n(z,w)|\le 1,|Q_n(z,w)|\le 1\}$ 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 ${\mathbf{C}}^2$, we construct a sequence of pairs $\{P_n,Q_n\}$ of homogeneous polynomials in two variables with $\mathrm{deg}{P}_n=$ $\mathrm{deg}{Q}_n=n$ such that the sets $K_n:=\{(z,w)\in {\mathbf{C}}^2:|P_n(z,w)|\le 1,|Q_n(z,w)|\le 1\}$ 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.

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