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Proper holomorphic mappings in several complex variables

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Title: Proper holomorphic mappings in several complex variables
Author(s): Setya-Budhi, Marcus Wono
Doctoral Committee Chair(s): Haboush, William J.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
Abstract: A holomorphic mapping f from a bounded domain D in $\doubc\sp{n}$ to a bounded domain $\Omega$ in $\doubc\sp{N}$ is proper if the sequence $\{f(zj)\}$ tends to the boundary of $\Omega$ for every sequence $\{zj\}$ which tends to the boundary of D. Let f be a proper holomorphic mapping from the unit ball in $\doubc\sp{n}$ to the unit ball in $\doubc\sp{N}$ for $N \ge n \ge 2.$ If f is a function of class $C\sp{N-n+1}$ on the closed unit ball and also satisfies a certain non-degeneracy condition, then Cima and Suffridge proved that f must be rational.In this thesis we prove a similar result for mappings from complex eggs to the unit ball in $\doubc\sp{N}.$ We also prove that a rational proper holomorphic mapping f from the complex egg $\{z \in \doubc\sp{n} \vert\Sigma\vert z\sb{i} < 1\}$ to the unit ball in $\doubc\sp{N} (N \ge n \ge 2)$ can be written as a composition H o g, where $H(z\sb1,\...,z\sb{n}) = (z\sbsp{1}{p1},\...,z\sbsp{n}{pn})$ and g is a proper holomorphic mapping from the unit ball in $\doubc\sp{n}$ to the unit ball in $\doubc\sp{N}.$The second part of this thesis concerns proper holomorphic rational mappings between balls in different dimensions. We give partial results about two conjectures. First, we prove that the degree of a proper holomorphic monomial mapping from $B\sb2$ to $B\sb5$ is at most 7. We also list all such examples. Second, we investigate the existence of a proper rational mapping P/q from the unit ball in $\doubc\sp{n}$ to the unit ball in $\doubc\sp{N},$ for certain allowable denominators q.
Issue Date: 1993
Type: Text
Language: English
URI: http://hdl.handle.net/2142/21458
Rights Information: Copyright 1993 Setya-Budhi, Marcus Wono
Date Available in IDEALS: 2011-05-07
Identifier in Online Catalog: AAI9411781
OCLC Identifier: (UMI)AAI9411781
 

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