{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98327"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98327","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Relative waring rank of binary forms","abstract":"This Dissertation was approved for publication on 2017-07-05 at 08:54.","abstract_html":"This Dissertation was approved for publication on 2017-07-05 at 08:54.","abstract_has_math":false,"creators":["Tokcan, Neriman"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Reznick, Bruce","Bergvelt, Maarten","Katz, Sheldon","Nevins, Thomas"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T17:56:21Z","date_published":"2017-09-29T17:56:21Z","updated_at":"2026-07-22T22:24:35Z","subjects":["Waring rank","Real rank","Binary forms","Sums of powers","Sylvester","Tensor decompositions"],"languages":["en"],"rights":["Copyright 2017 Neriman Tokcan"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98327","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Reznick, Bruce","Bergvelt, Maarten","Katz, Sheldon","Nevins, Thomas"]},{"key":"dc:creator","label":"Author","values":["Tokcan, Neriman"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-09-29T17:56:21Z","2017-07-05","2017-08"]},{"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":["Waring rank","Real rank","Binary forms","Sums of powers","Sylvester","Tensor decompositions"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Neriman Tokcan"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98327"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This Dissertation was approved for publication on 2017-07-05 at 08:54.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11304 on 2017-09-29 at 11:27:48","Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \\subseteq \\cc$. The {\\it $K$-rank of $f$} is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d \\ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fields. We also study the relation between the relative rank and the algebraic properties of the underlying field. In particular, we show that $K$-rank of a form $f$ (such as $x^3y^2$) may depend on whether $-1$ is a sum of two squares in $K.$ We provide lower bounds for the $\\mathbb{C}$-rank (Waring rank) and for the $\\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We also give the rank of quartic and quintic binary forms based on their factorization over $\\cc.$ We investigate the structure of binary forms with unique $\\mathbb{C}$-minimal representation.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Neriman Tokcan, accepted the attached license on 2017-07-03 at 13:33.","The student, Neriman Tokcan, submitted this Dissertation for approval on 2017-07-03 at 14:01.","Made available in DSpace on 2017-09-29T17:56:21Z (GMT). No. of bitstreams: 3 TOKCAN-DISSERTATION-2017.pdf: 387608 bytes, checksum: 0bdce167a835000aff6764393bea5cc1 (MD5) Tokcan thesis.tex: 133929 bytes, checksum: 6bba2f4fb581b218058fa4b863dc4876 (MD5) LICENSE.txt: 4211 bytes, checksum: 81aed8dbdf5297c80c0bbc81b3234f08 (MD5) Previous issue date: 2017-07-05"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Relative waring rank of binary forms"]}]}],"canonical_facts":{"dc:contributor":["Reznick, Bruce","Bergvelt, Maarten","Katz, Sheldon","Nevins, Thomas"],"dc:creator":["Tokcan, Neriman"],"dc:date":["2017-09-29T17:56:21Z","2017-07-05","2017-08"],"dc:description":["This Dissertation was approved for publication on 2017-07-05 at 08:54.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11304 on 2017-09-29 at 11:27:48","Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \\subseteq \\cc$. The {\\it $K$-rank of $f$} is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d \\ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fields. We also study the relation between the relative rank and the algebraic properties of the underlying field. In particular, we show that $K$-rank of a form $f$ (such as $x^3y^2$) may depend on whether $-1$ is a sum of two squares in $K.$ We provide lower bounds for the $\\mathbb{C}$-rank (Waring rank) and for the $\\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We also give the rank of quartic and quintic binary forms based on their factorization over $\\cc.$ We investigate the structure of binary forms with unique $\\mathbb{C}$-minimal representation.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Neriman Tokcan, accepted the attached license on 2017-07-03 at 13:33.","The student, Neriman Tokcan, submitted this Dissertation for approval on 2017-07-03 at 14:01.","Made available in DSpace on 2017-09-29T17:56:21Z (GMT). No. of bitstreams: 3 TOKCAN-DISSERTATION-2017.pdf: 387608 bytes, checksum: 0bdce167a835000aff6764393bea5cc1 (MD5) Tokcan thesis.tex: 133929 bytes, checksum: 6bba2f4fb581b218058fa4b863dc4876 (MD5) LICENSE.txt: 4211 bytes, checksum: 81aed8dbdf5297c80c0bbc81b3234f08 (MD5) Previous issue date: 2017-07-05"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/98327"],"dc:language":["en"],"dc:rights":["Copyright 2017 Neriman Tokcan"],"dc:subject":["Waring rank","Real rank","Binary forms","Sums of powers","Sylvester","Tensor decompositions"],"dc:title":["Relative waring rank of binary forms"],"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":"2026-07-22T22:24:35Z"}