{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86784"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86784","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Function Field Arithmetic and Related Algorithms","abstract":"The second part of the thesis is focussed on developing an explicit arithmetic for the Jacobian of certain cubic superelliptic curves. We restrict our attention to curves of the form y3 = f( x). Assuming that f(x) is monic with no repeated roots and that our field does not have characteristic 3, we are able to show that the Jacobian of this curve is isomorphic to the ideal class group of K[C], the ring of regular functions on C. By exploiting the structure of ideals in K[C] as K[x] modules, we are able to produce a very efficient algorithm for performing group operations in the Jacobian which heuristically should take 46g 2 operations in the finite field K.","abstract_html":"The second part of the thesis is focussed on developing an explicit arithmetic for the Jacobian of certain cubic superelliptic curves. We restrict our attention to curves of the form y3 = f( x). Assuming that f(x) is monic with no repeated roots and that our field does not have characteristic 3, we are able to show that the Jacobian of this curve is isomorphic to the ideal class group of K[C], the ring of regular functions on C. By exploiting the structure of ideals in K[C] as K[x] modules, we are able to produce a very efficient algorithm for performing group operations in the Jacobian which heuristically should take 46g 2 operations in the finite field K.","abstract_has_math":false,"creators":["Bauer, Mark L."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Boston, Nigel"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:31Z","date_published":"2015-09-28T15:19:31Z","updated_at":"2026-07-22T22:26:27Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3023016"],"render_values":[{"text":"(MiAaPQ)AAI3023016","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86784","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Boston, Nigel"]},{"key":"dc:creator","label":"Author","values":["Bauer, Mark L."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:31Z","10000-01-01","2001"]},{"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/2142/86784","(MiAaPQ)AAI3023016"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The second part of the thesis is focussed on developing an explicit arithmetic for the Jacobian of certain cubic superelliptic curves. 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Assuming that f(x) is monic with no repeated roots and that our field does not have characteristic 3, we are able to show that the Jacobian of this curve is isomorphic to the ideal class group of K[C], the ring of regular functions on C. By exploiting the structure of ideals in K[C] as K[x] modules, we are able to produce a very efficient algorithm for performing group operations in the Jacobian which heuristically should take 46g 2 operations in the finite field K.","Made available in DSpace on 2015-09-28T15:19:31Z (GMT). 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