[Formula: see text] -Functions in Unbounded Balanced Domains

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 [Formula: see text] -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 [Formula: see text] . This allows easily to decide which pseudoconvex balanced domain in [Formula: see text] has a positive Bergman kernel and which admits the Bergman metric.