{"id":{"repo_id":"ubc","oai_identifier":"oai:circle.library.ubc.ca:2429/1288"},"canonical_url":"https://search.dev.ndltd.org/etd/ubc/oai:circle.library.ubc.ca:2429/1288","repository":{"repo_id":"ubc","name":"University of British Columbia","base_url":"http://circle.library.ubc.ca/oai/request"},"display":{"title":"Thue equations and related topics","abstract":"Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas.","abstract_html":"Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas.","abstract_has_math":false,"creators":["Akhtari, Shabnam"],"institution":"University of British Columbia","degree_name":"Doctor of Philosophy - PhD","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008","date_published":"2008","updated_at":"2026-07-24T05:07:35Z","subjects":[],"languages":["eng"],"rights":["Attribution-NonCommercial-NoDerivatives 4.0 International"],"rights_urls":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2429/1288","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Akhtari, Shabnam"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2008"]},{"key":"dc:publisher","label":"Institution","values":["University of British Columbia"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy - PhD"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of British Columbia"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["http://creativecommons.org/licenses/by-nc-nd/4.0/","Attribution-NonCommercial-NoDerivatives 4.0 International"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2429/1288","http://circle.library.ubc.ca/bitstream/2429/1288/1/ubc_2008_fall_akhtari_shabnam.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas."]},{"key":"dc:format","label":"Dc Format","values":["768692","application/pdf"]},{"key":"dc:title","label":"Title","values":["Thue equations and related topics"]}]}],"canonical_facts":{"dc:creator":["Akhtari, Shabnam"],"dc:date":["2008"],"dc:description":["Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas."],"dc:format":["768692","application/pdf"],"dc:identifier":["http://hdl.handle.net/2429/1288","http://circle.library.ubc.ca/bitstream/2429/1288/1/ubc_2008_fall_akhtari_shabnam.pdf"],"dc:language":["eng"],"dc:publisher":["University of British Columbia"],"dc:rights":["http://creativecommons.org/licenses/by-nc-nd/4.0/","Attribution-NonCommercial-NoDerivatives 4.0 International"],"dc:title":["Thue equations and related topics"],"dc:type":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy - PhD"],"thesis:institution_name":["University of British Columbia"]},"updated_at":"2026-07-24T05:07:35Z"}