Proper holomorphic embeddings, radial approximation, and tropical power series

We study the problem of approximation of radial weights by finite sums of moduli of holomorphic functions. Relations to other approximation problems are discussed. As an application, quantitative aspects of proper holomorphic embeddings of planar domains and complex balls into $\mathbb C^N$ are considered.
The talk is based on joint work with E. Dubtsov.