Growth of proper holomorphic maps and tropical power series

Evgeny Abakumov; Evgueni Doubtsov

doi:10.1016/j.crma.2016.03.001

Complex analysis

Growth of proper holomorphic maps and tropical power series
[Croissance d'applications holomorphes propres et séries entières tropicales]

Présenté par : Jean-Pierre Demailly

Evgeny Abakumov ¹ ; Evgueni Doubtsov ^2,³

¹ Université Paris-Est, LAMA (UMR 8050), 77454 Marne-la-Vallée, France
² St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
³ Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr. 28, St. Petersburg 198504, Russia

Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 465-469.

Résumé
Abstract

Motivés par une question de J. Globevnik, nous montrons qu'une immersion holomorphe propre du disque unité $D$ dans $C^{2}$ ou un plongement holomorphe propre $f : D \to C^{3}$ peut avoir une croissance arbitraire. En outre, en utilisant les séries entières tropicales, nous caractérisons les poids radiaux w sur le plan complexe pour lesquels il existe $n \in N$ et une application holomorphe propre $f : C \to C^{n}$ tels que $| f (z) |$ soit équivalente à $w (z)$ .

Motivated by a question of J. Globevnik, we show that a proper holomorphic immersion of the unit disk $D$ into $C^{2}$ or a proper holomorphic embedding $f : D \to C^{3}$ may have arbitrary growth. Also, using tropical power series, we characterize those radial weights w on the complex plane for which there exist $n \in N$ and a proper holomorphic map $f : C \to C^{n}$ such that $| f (z) |$ is equivalent to $w (z)$ .

Reçu le : 2016-01-12
Accepté le : 2016-03-03
Publié le : 2016-03-29

DOI : 10.1016/j.crma.2016.03.001

Affiliations des auteurs :

Evgeny Abakumov ¹ ; Evgueni Doubtsov ^{2, 3}

¹ Université Paris-Est, LAMA (UMR 8050), 77454 Marne-la-Vallée, France
² St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
³ Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr. 28, St. Petersburg 198504, Russia

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     author = {Evgeny Abakumov and Evgueni Doubtsov},
     title = {Growth of proper holomorphic maps and tropical power series},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {465--469},
     publisher = {Elsevier},
     volume = {354},
     number = {5},
     year = {2016},
     doi = {10.1016/j.crma.2016.03.001},
     language = {en},
}

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Evgeny Abakumov; Evgueni Doubtsov. Growth of proper holomorphic maps and tropical power series. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 465-469. doi : 10.1016/j.crma.2016.03.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.03.001/

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