Abstract
Motivated by a question of J. Globevnik, we show that a proper holomorphic immersion of the unit disk D into C2 or a proper holomorphic embedding f:D→C3 may have arbitrary growth. Also, using tropical power series, we characterize those radial weights w on the complex plane for which there exist n∈N and a proper holomorphic map f:C→Cn such that |f(z)| is equivalent to w(z).
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