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arXiv · 1510.01616

Approximation by proper holomorphic maps and tropical power series

Evgeny Abakumov·Evgueni Doubtsov
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Abstract

Let $w$ be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map $f: \mathbb{C}\to\mathbb{C}^n$ such that $|f|$ is equivalent to $w$. We give several characterizations of those $w$ for which the problem is solvable. In particular, a constructive characterization is given in terms of tropical power series. Moreover, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, approximation by the maximum of a holomorphic function modulus. Extensions to several complex variables and approximation by harmonic maps are also considered.

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Evgeny Abakumov, Evgueni Doubtsov. 2015-10-06. Approximation by proper holomorphic maps and tropical power series. https://arxiv.org/abs/1510.01616

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