{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21458"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21458","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Proper holomorphic mappings in several complex variables","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.","abstract_html":"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.","abstract_has_math":true,"creators":["Setya-Budhi, Marcus Wono"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Haboush, William J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:09:13Z","date_published":"2011-05-07T13:09:13Z","updated_at":"2026-07-22T22:25:18Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1993 Setya-Budhi, Marcus Wono"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9411781","(UMI)AAI9411781"],"render_values":[{"text":"AAI9411781","href":null,"code":true},{"text":"(UMI)AAI9411781","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21458","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Haboush, William J."]},{"key":"dc:creator","label":"Author","values":["Setya-Budhi, Marcus Wono"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:09:13Z","10000-01-01","1993"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1993 Setya-Budhi, Marcus Wono"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9411781","(UMI)AAI9411781","http://hdl.handle.net/2142/21458"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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"]},{"key":"dc:title","label":"Title","values":["Proper holomorphic mappings in several complex variables"]}]}],"canonical_facts":{"dc:contributor":["Haboush, William J."],"dc:creator":["Setya-Budhi, Marcus Wono"],"dc:date":["2011-05-07T13:09:13Z","10000-01-01","1993"],"dc:description":["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"],"dc:identifier":["AAI9411781","(UMI)AAI9411781","http://hdl.handle.net/2142/21458"],"dc:language":["eng"],"dc:rights":["Copyright 1993 Setya-Budhi, Marcus Wono"],"dc:subject":["Mathematics"],"dc:title":["Proper holomorphic mappings in several complex variables"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:18Z"}