{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86922"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86922","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5","abstract":"Finally, we describe methods to compute the divisor class number, h, of K, and in the case that O has unit rank 1 or 2, the regulator and ideal class number of O as well. A method of Scheidler and Stein [SS07, SS08] determines sharper upper and lower bounds on h, for a given cubic function field, than those given by the Hasse-Weil Theorem. We then employ Shanks' Baby Step-Giant Step algorithm [Sha71] and Pollard's Kangaroo method [Pol78], to search this interval and compute the desired invariants for purely cubic function fields of unit rank 0 and 1. The total complexity of the method to compute these invariants is O (q(2 g-1)/5+epsilon(g )) ideal operations as q → infinity, where 0 ≤ epsilon( g) ≤ 1/5. With this approach, we computed the 28 decimal digit divisor class numbers of two purely cubic function fields of genus 3: one of unit rank 0 and one of unit rank 1. We also computed the 25 decimal digit divisor class numbers of two purely cubic function fields of genus 4: one of unit rank 0 and one of unit rank 1. In the unit rank 1 examples, we factored the divisor class numbers into the ideal class numbers and the respective 26 and 24 decimal digit S-regulators. We believe that these are the largest divisor class numbers ever computed for a cubic function field of genus at least 4 and the largest regulators ever computed for any cubic function field, respectively.","abstract_html":"Finally, we describe methods to compute the divisor class number, h, of K, and in the case that O has unit rank 1 or 2, the regulator and ideal class number of O as well. A method of Scheidler and Stein [SS07, SS08] determines sharper upper and lower bounds on h, for a given cubic function field, than those given by the Hasse-Weil Theorem. We then employ Shanks' Baby Step-Giant Step algorithm [Sha71] and Pollard's Kangaroo method [Pol78], to search this interval and compute the desired invariants for purely cubic function fields of unit rank 0 and 1. The total complexity of the method to compute these invariants is O (q(2 g-1)/5+epsilon(g )) ideal operations as q → infinity, where 0 ≤ epsilon( g) ≤ 1/5. With this approach, we computed the 28 decimal digit divisor class numbers of two purely cubic function fields of genus 3: one of unit rank 0 and one of unit rank 1. We also computed the 25 decimal digit divisor class numbers of two purely cubic function fields of genus 4: one of unit rank 0 and one of unit rank 1. In the unit rank 1 examples, we factored the divisor class numbers into the ideal class numbers and the respective 26 and 24 decimal digit S-regulators. We believe that these are the largest divisor class numbers ever computed for a cubic function field of genus at least 4 and the largest regulators ever computed for any cubic function field, respectively.","abstract_has_math":false,"creators":["Landquist, Eric"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Zaharescu, Alexandru"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:11Z","date_published":"2015-09-28T15:20:11Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3363008"],"render_values":[{"text":"(MiAaPQ)AAI3363008","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86922","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zaharescu, Alexandru"]},{"key":"dc:creator","label":"Author","values":["Landquist, Eric"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:11Z","10000-01-01","2009"]},{"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/2142/86922","(MiAaPQ)AAI3363008"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Finally, we describe methods to compute the divisor class number, h, of K, and in the case that O has unit rank 1 or 2, the regulator and ideal class number of O as well. A method of Scheidler and Stein [SS07, SS08] determines sharper upper and lower bounds on h, for a given cubic function field, than those given by the Hasse-Weil Theorem. We then employ Shanks' Baby Step-Giant Step algorithm [Sha71] and Pollard's Kangaroo method [Pol78], to search this interval and compute the desired invariants for purely cubic function fields of unit rank 0 and 1. The total complexity of the method to compute these invariants is O (q(2 g-1)/5+epsilon(g )) ideal operations as q → infinity, where 0 ≤ epsilon( g) ≤ 1/5. With this approach, we computed the 28 decimal digit divisor class numbers of two purely cubic function fields of genus 3: one of unit rank 0 and one of unit rank 1. We also computed the 25 decimal digit divisor class numbers of two purely cubic function fields of genus 4: one of unit rank 0 and one of unit rank 1. In the unit rank 1 examples, we factored the divisor class numbers into the ideal class numbers and the respective 26 and 24 decimal digit S-regulators. We believe that these are the largest divisor class numbers ever computed for a cubic function field of genus at least 4 and the largest regulators ever computed for any cubic function field, respectively.","Made available in DSpace on 2015-09-28T15:20:11Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3363008.pdf: 3926833 bytes, checksum: 7ed0ec2cbf9e227cdcd4dda8073165c8 (MD5) Previous issue date: 2009","Embargo set by: Seth Robbins for item 88203 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","194 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2009."]},{"key":"dc:title","label":"Title","values":["Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5"]}]}],"canonical_facts":{"dc:contributor":["Zaharescu, Alexandru"],"dc:creator":["Landquist, Eric"],"dc:date":["2015-09-28T15:20:11Z","10000-01-01","2009"],"dc:description":["Finally, we describe methods to compute the divisor class number, h, of K, and in the case that O has unit rank 1 or 2, the regulator and ideal class number of O as well. A method of Scheidler and Stein [SS07, SS08] determines sharper upper and lower bounds on h, for a given cubic function field, than those given by the Hasse-Weil Theorem. We then employ Shanks' Baby Step-Giant Step algorithm [Sha71] and Pollard's Kangaroo method [Pol78], to search this interval and compute the desired invariants for purely cubic function fields of unit rank 0 and 1. The total complexity of the method to compute these invariants is O (q(2 g-1)/5+epsilon(g )) ideal operations as q → infinity, where 0 ≤ epsilon( g) ≤ 1/5. With this approach, we computed the 28 decimal digit divisor class numbers of two purely cubic function fields of genus 3: one of unit rank 0 and one of unit rank 1. We also computed the 25 decimal digit divisor class numbers of two purely cubic function fields of genus 4: one of unit rank 0 and one of unit rank 1. In the unit rank 1 examples, we factored the divisor class numbers into the ideal class numbers and the respective 26 and 24 decimal digit S-regulators. We believe that these are the largest divisor class numbers ever computed for a cubic function field of genus at least 4 and the largest regulators ever computed for any cubic function field, respectively.","Made available in DSpace on 2015-09-28T15:20:11Z (GMT). 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