Approximation by proper holomorphic maps and tropical power series

TL;DR

This paper investigates conditions under which unbounded radial weights on the complex plane can be approximated by proper holomorphic maps, introducing tropical power series as a key tool and extending results to multiple complex variables and harmonic maps.

Contribution

It provides a constructive characterization of weights allowing such approximation using tropical power series, advancing understanding of approximation by proper holomorphic maps.

Findings

Characterization of weights solvable by proper holomorphic maps

Use of tropical power series for constructive approximation

Extensions to several complex variables and harmonic maps

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.