A Hilbert Lemniscate Theorem in C^2

Details A Hilbert Lemniscate Theorem in C^2 A Hilbert Lemniscate Theorem in C^2

2006-07-22

arxiv.org/abs/math/0607574v1

Mathematics

Complex Variables

Scientific paper

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) \in C^2 : |P_n(z,w)| \leq 1, |Q_n(z,w)| \leq 1} approximate K and the normalized counting measures {\mu_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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