A Hilbert Lemniscate Theorem in C^2
Complex Variables
2007-05-23 v1
Abstract
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.
Cite
@article{arxiv.math/0607574,
title = {A Hilbert Lemniscate Theorem in C^2},
author = {T. Bloom and N. Levenberg and Yu. Lyubarskii},
journal= {arXiv preprint arXiv:math/0607574},
year = {2007}
}