{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72784"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72784","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Ideals of powers of linear forms","abstract":"This thesis addresses two closely related problems about ideals of powers of linear forms. In the first chapter, we analyze a problem from spline theory, namely to compute the dimension of the vector space of tri-variate splines on a special class of tetrahedral complexes, using ideals of powers of linear forms. By Macaulay's inverse system, this class of ideals is closely related to ideals of fat points. In the second chapter, we approach a conjecture of Postnikov and Shapiro concerning the minimal free resolutions of a class of ideals of powers of linear forms in n variables which are constructed from complete graphs on n + 1 vertices. This statement was also conjectured by Schenck in the special case of n = 3. We provide two different approaches to his conjecture. We prove the conjecture of Postnikov and Shapiro under the additional condition that certain modules are free.","abstract_html":"This thesis addresses two closely related problems about ideals of powers of linear forms. In the first chapter, we analyze a problem from spline theory, namely to compute the dimension of the vector space of tri-variate splines on a special class of tetrahedral complexes, using ideals of powers of linear forms. By Macaulay's inverse system, this class of ideals is closely related to ideals of fat points. In the second chapter, we approach a conjecture of Postnikov and Shapiro concerning the minimal free resolutions of a class of ideals of powers of linear forms in n variables which are constructed from complete graphs on n + 1 vertices. This statement was also conjectured by Schenck in the special case of n = 3. We provide two different approaches to his conjecture. We prove the conjecture of Postnikov and Shapiro under the additional condition that certain modules are free.","abstract_has_math":false,"creators":["Shan, Jianyun"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Schenck, Hal","D'Angelo, John","Nevins, Thomas A.","Yong, Alexander"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-01-21T19:48:10Z","date_published":"2015-01-21T19:48:10Z","updated_at":"2026-07-22T22:26:07Z","subjects":["Splines","fat points","free resolutions","powers of linear forms"],"languages":["en"],"rights":["Copyright 2014 Jianyun Shan"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/72784","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Schenck, Hal","D'Angelo, John","Nevins, Thomas A.","Yong, Alexander"]},{"key":"dc:creator","label":"Author","values":["Shan, Jianyun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-01-21T19:48:10Z","2014-12","2015-01-21"]},{"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":["Splines","fat points","free resolutions","powers of linear forms"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Jianyun Shan"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/72784"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis addresses two closely related problems about ideals of powers of linear forms. 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