A Hilbert Lemniscate Theorem in C2
T. Bloom, N. Levenberg, Yu. Lyubarskii
Abstract
For a regular, compact, polynomially convex circled set K in C2, we construct a sequence of pairs Pn,Qn of homogeneous polynomials in two variables with deg Pn = deg Qn = n such that the sets Kn: = (z,w) ∈ C2 : |Pn(z,w)| ≤ 1, |Qn(z,w)| ≤ 1 approximate K and the normalized counting measures μn associated to the finite set Pn = Qn = 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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