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Title:  Proper holomorphic mappings in several complex variables 
Author(s):  SetyaBudhi, Marcus Wono 
Doctoral Committee Chair(s):  Haboush, William J. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
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 $C\sp{Nn+1}$ on the closed unit ball and also satisfies a certain nondegeneracy 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 SetyaBudhi, Marcus Wono 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9411781 
OCLC Identifier:  (UMI)AAI9411781 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics