{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/97285"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/97285","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Syzygies and implicitization of tensor product surfaces","abstract":"A tensor product surface is the closure of the image of a rational map λ : P1 ×P1-->P3. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of λ in P3. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map λ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of λ are closely related to the geometry of the set of points at which λ is undefined.","abstract_html":"A tensor product surface is the closure of the image of a rational map λ : P1 ×P1-->P3. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of λ in P3. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map λ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of λ are closely related to the geometry of the set of points at which λ is undefined.","abstract_has_math":false,"creators":["Duarte Gelvez, Eliana Maria"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Schenck, Henry","Nevins, Thomas","Francis, George","Reznick, Bruce"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-08-10T19:14:37Z","date_published":"2017-08-10T19:14:37Z","updated_at":"2026-07-22T22:24:32Z","subjects":["Implicitization","Syzygy","Rees algebras","Basepoints","Tensor product surface","Smooth quadric"],"languages":["en"],"rights":["Copyright 2017 Eliana Duarte"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/97285","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Schenck, Henry","Nevins, Thomas","Francis, George","Reznick, Bruce"]},{"key":"dc:creator","label":"Author","values":["Duarte Gelvez, Eliana Maria"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-08-10T19:14:37Z","2017-04-10","2017-05"]},{"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":["Implicitization","Syzygy","Rees algebras","Basepoints","Tensor product surface","Smooth quadric"]}]},{"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 Eliana Duarte"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/97285"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A tensor product surface is the closure of the image of a rational map λ : P1 ×P1-->P3. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of λ in P3. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map λ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of λ are closely related to the geometry of the set of points at which λ is undefined.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Eliana Duarte Gelvez, accepted the attached license on 2017-03-23 at 04:01.","The student, Eliana Duarte Gelvez, submitted this Dissertation for approval on 2017-03-23 at 04:13.","This Dissertation was approved for publication on 2017-04-10 at 08:45.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10612 on 2017-08-10 at 13:38:20","Made available in DSpace on 2017-08-10T19:14:37Z (GMT). No. of bitstreams: 2 DUARTEGELVEZ-DISSERTATION-2017.pdf: 752902 bytes, checksum: 5f8ce797b270f9e39cbd70f9112bf252 (MD5) LICENSE.txt: 4217 bytes, checksum: 5dce0ef5c5f24c89bc8021d82954464b (MD5) Previous issue date: 2017-04-10"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Syzygies and implicitization of tensor product surfaces"]}]}],"canonical_facts":{"dc:contributor":["Schenck, Henry","Nevins, Thomas","Francis, George","Reznick, Bruce"],"dc:creator":["Duarte Gelvez, Eliana Maria"],"dc:date":["2017-08-10T19:14:37Z","2017-04-10","2017-05"],"dc:description":["A tensor product surface is the closure of the image of a rational map λ : P1 ×P1-->P3. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of λ in P3. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map λ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of λ are closely related to the geometry of the set of points at which λ is undefined.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Eliana Duarte Gelvez, accepted the attached license on 2017-03-23 at 04:01.","The student, Eliana Duarte Gelvez, submitted this Dissertation for approval on 2017-03-23 at 04:13.","This Dissertation was approved for publication on 2017-04-10 at 08:45.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10612 on 2017-08-10 at 13:38:20","Made available in DSpace on 2017-08-10T19:14:37Z (GMT). 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