{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86812"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86812","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Waring's Problem for Linear Polynomials and Laurent Polynomials","abstract":"Waring's problem is about representing any function in a class of functions as a sum of kth powers of nonconstant functions in the same class. We allow complex coefficients in these kind of problems. Consider i=1p1 fiz k=z and i=1p1 fizk =1 . For a given k ≥ 2, let p 1 and p2 be the smallest numbers of functions that give the above identities. W. K. Hayman obtained lower bounds of p1 and p2 for polynomials, entire functions, rational functions and meromorphic functions. First, we consider Waring's problem for linear polynomials and get p 1 = k and p2 ≥ k + 1. Next, we study Waring's problem for Laurent polynomials and obtain lower bounds of p1 and p 2. Finally, we discuss the misquote that I discovered in the proof of Hayman's theorem.","abstract_html":"Waring's problem is about representing any function in a class of functions as a sum of kth powers of nonconstant functions in the same class. We allow complex coefficients in these kind of problems. Consider i=1p1 fiz k=z and i=1p1 fizk =1 . For a given k ≥ 2, let p 1 and p2 be the smallest numbers of functions that give the above identities. W. K. Hayman obtained lower bounds of p1 and p2 for polynomials, entire functions, rational functions and meromorphic functions. First, we consider Waring's problem for linear polynomials and get p 1 = k and p2 ≥ k + 1. Next, we study Waring's problem for Laurent polynomials and obtain lower bounds of p1 and p 2. Finally, we discuss the misquote that I discovered in the proof of Hayman's theorem.","abstract_has_math":false,"creators":["Kim, Dong-Il"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Aimo Hinkkanen"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:39Z","date_published":"2015-09-28T15:19:39Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3086095"],"render_values":[{"text":"(MiAaPQ)AAI3086095","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86812","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Aimo Hinkkanen"]},{"key":"dc:creator","label":"Author","values":["Kim, Dong-Il"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:39Z","10000-01-01","2003"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86812","(MiAaPQ)AAI3086095"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Waring's problem is about representing any function in a class of functions as a sum of kth powers of nonconstant functions in the same class. We allow complex coefficients in these kind of problems. Consider i=1p1 fiz k=z and i=1p1 fizk =1 . For a given k ≥ 2, let p 1 and p2 be the smallest numbers of functions that give the above identities. W. K. Hayman obtained lower bounds of p1 and p2 for polynomials, entire functions, rational functions and meromorphic functions. First, we consider Waring's problem for linear polynomials and get p 1 = k and p2 ≥ k + 1. Next, we study Waring's problem for Laurent polynomials and obtain lower bounds of p1 and p 2. Finally, we discuss the misquote that I discovered in the proof of Hayman's theorem.","Made available in DSpace on 2015-09-28T15:19:39Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3086095.pdf: 1883591 bytes, checksum: 50a60b280531d3b94a80505beecbca97 (MD5) Previous issue date: 2003","Embargo set by: Seth Robbins for item 88093 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","52 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003."]},{"key":"dc:title","label":"Title","values":["Waring's Problem for Linear Polynomials and Laurent Polynomials"]}]}],"canonical_facts":{"dc:contributor":["Aimo Hinkkanen"],"dc:creator":["Kim, Dong-Il"],"dc:date":["2015-09-28T15:19:39Z","10000-01-01","2003"],"dc:description":["Waring's problem is about representing any function in a class of functions as a sum of kth powers of nonconstant functions in the same class. We allow complex coefficients in these kind of problems. Consider i=1p1 fiz k=z and i=1p1 fizk =1 . For a given k ≥ 2, let p 1 and p2 be the smallest numbers of functions that give the above identities. W. K. Hayman obtained lower bounds of p1 and p2 for polynomials, entire functions, rational functions and meromorphic functions. First, we consider Waring's problem for linear polynomials and get p 1 = k and p2 ≥ k + 1. Next, we study Waring's problem for Laurent polynomials and obtain lower bounds of p1 and p 2. Finally, we discuss the misquote that I discovered in the proof of Hayman's theorem.","Made available in DSpace on 2015-09-28T15:19:39Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3086095.pdf: 1883591 bytes, checksum: 50a60b280531d3b94a80505beecbca97 (MD5) Previous issue date: 2003","Embargo set by: Seth Robbins for item 88093 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","52 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003."],"dc:identifier":["http://hdl.handle.net/2142/86812","(MiAaPQ)AAI3086095"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Waring's Problem for Linear Polynomials and Laurent Polynomials"],"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":"2026-07-22T22:26:28Z"}