{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/44453"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/44453","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Root distribution of polynomials and distance sums on the unit circle","abstract":"Our first topic is the study of self-inversive polynomials. We establish sufficient conditions for self-inversive polynomials to have all zeros on the unit circle. We examine how such polynomials are used to generate further such polynomials. We also analyze the distribution of zeros of certain families of self-inversive polynomials and evaluate their discriminants. Our second topic is the sum of distances between points on the unit circle. We consider the sums over the vertices of the regular $N$-gon with some stretching factors. We also consider the $N$ vertices of a regular $N$-gon with charges on the unit circle and obtain the maximal sum of squared distances from a point on the unit circle to the charged $N$-gon. For a certain set of charges, we study the minimal polynomial of the maximum value and its generating function.","abstract_html":"Our first topic is the study of self-inversive polynomials. We establish sufficient conditions for self-inversive polynomials to have all zeros on the unit circle. We examine how such polynomials are used to generate further such polynomials. We also analyze the distribution of zeros of certain families of self-inversive polynomials and evaluate their discriminants. Our second topic is the sum of distances between points on the unit circle. We consider the sums over the vertices of the regular $N$-gon with some stretching factors. We also consider the $N$ vertices of a regular $N$-gon with charges on the unit circle and obtain the maximal sum of squared distances from a point on the unit circle to the charged $N$-gon. For a certain set of charges, we study the minimal polynomial of the maximum value and its generating function.","abstract_has_math":true,"creators":["Kim, Eunmi"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Stolarsky, Kenneth B.","Hildebrand, A.J.","Berndt, Bruce C.","Zaharescu, Alexandru"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-24T22:16:39Z","date_published":"2013-05-24T22:16:39Z","updated_at":"2026-07-22T22:25:34Z","subjects":["self-inversive polynomials","distance sum","unit circle"],"languages":["en"],"rights":["Copyright 2013 Eunmi Kim"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/44453","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Stolarsky, Kenneth B.","Hildebrand, A.J.","Berndt, Bruce C.","Zaharescu, Alexandru"]},{"key":"dc:creator","label":"Author","values":["Kim, Eunmi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-05-24T22:16:39Z","2015-05-24T10:01:49Z","2013-05"]},{"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":["self-inversive polynomials","distance sum","unit circle"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2013 Eunmi Kim"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/44453"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Our first topic is the study of self-inversive polynomials. We establish sufficient conditions for self-inversive polynomials to have all zeros on the unit circle. We examine how such polynomials are used to generate further such polynomials. We also analyze the distribution of zeros of certain families of self-inversive polynomials and evaluate their discriminants. Our second topic is the sum of distances between points on the unit circle. We consider the sums over the vertices of the regular $N$-gon with some stretching factors. We also consider the $N$ vertices of a regular $N$-gon with charges on the unit circle and obtain the maximal sum of squared distances from a point on the unit circle to the charged $N$-gon. For a certain set of charges, we study the minimal polynomial of the maximum value and its generating function.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-03-20T13:35:28Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Kim_Eunmi.pdf: 727585 bytes, checksum: c061cf15a75e8d40c93a9e1cf75158bd (MD5)","Made available in DSpace on 2013-05-24T22:16:39Z (GMT). No. of bitstreams: 2 Eunmi_Kim.pdf: 727585 bytes, checksum: c061cf15a75e8d40c93a9e1cf75158bd (MD5) license.txt: 4057 bytes, checksum: 2d80322c707f9e95457be352e96b4869 (MD5)","Restriction data tranferred 2014-07-01T11:36:13-05:00 Original Data Group with Access UIUC Users [automated] Release Date: 2015-05-24 17:18:31 UTC Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Seth Robbins (srobbins@illinois.edu) on 2013-05-24T22:19:05Z Item is restricted until 2015-05-24T22:18:31Z","U of I Only Restriction Lifted for Item 44426 on 2015-05-24T10:01:49Z."]},{"key":"dc:title","label":"Title","values":["Root distribution of polynomials and distance sums on the unit circle"]}]}],"canonical_facts":{"dc:contributor":["Stolarsky, Kenneth B.","Hildebrand, A.J.","Berndt, Bruce C.","Zaharescu, Alexandru"],"dc:creator":["Kim, Eunmi"],"dc:date":["2013-05-24T22:16:39Z","2015-05-24T10:01:49Z","2013-05"],"dc:description":["Our first topic is the study of self-inversive polynomials. We establish sufficient conditions for self-inversive polynomials to have all zeros on the unit circle. We examine how such polynomials are used to generate further such polynomials. We also analyze the distribution of zeros of certain families of self-inversive polynomials and evaluate their discriminants. Our second topic is the sum of distances between points on the unit circle. We consider the sums over the vertices of the regular $N$-gon with some stretching factors. We also consider the $N$ vertices of a regular $N$-gon with charges on the unit circle and obtain the maximal sum of squared distances from a point on the unit circle to the charged $N$-gon. For a certain set of charges, we study the minimal polynomial of the maximum value and its generating function.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-03-20T13:35:28Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Kim_Eunmi.pdf: 727585 bytes, checksum: c061cf15a75e8d40c93a9e1cf75158bd (MD5)","Made available in DSpace on 2013-05-24T22:16:39Z (GMT). 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