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Haboush, William J. Setya-Budhi, Marcus Wono 2011-05-07T13:09:13Z 2011-05-07T13:09:13Z 10000-01-01 1993 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. Made available in DSpace on 2011-05-07T13:09:13Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9411781.pdf: 2033300 bytes, checksum: 93a8d6606ae0641c43c6289ff95d62d1 (MD5) Previous issue date: 1993 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:50:55Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:23:19-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9411781 (UMI)AAI9411781 http://hdl.handle.net/2142/21458 eng Copyright 1993 Setya-Budhi, Marcus Wono Mathematics Proper holomorphic mappings in several complex variables text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.