A Hilbert Lemniscate Theorem in C²

2006/07/22 by Thomas Bloom, Bloom, T., N. Levenberg +3 · 1 citation
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.math/0607574

Abstract

For a regular, compact, polynomially convex circled set K in C², 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) ∈ C² : |P_n(z,w)| ≤ 1, |Q_n(z,w)| ≤ 1 approximate K and the normalized counting measures μ_n associated to the finite set P_n = Q_n = 1 converge to the pluripotential-theoretic Monge-Ampere measure for K. The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex variable.

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