L_h^2-Functions in Unbounded Balanced Domains

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We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a description of L_h^2-domains of holomorphy in the class of balanced domains and present a purely algebraic criterion for homogeneous polynomials to be square integrable in a pseudoconvex balanced domain in mathbb {C}^2. This allows easily to decide which pseudoconvex balanced domain in mathbb {C}^2 has a positive Bergman kernel and which admits the Bergman metric.