{"id":"https://openalex.org/W7204171943","doi":"https://doi.org/10.48550/arxiv.2608.23559","title":"Power-saving bounds for Thue--Mahler and Mordell equations","display_name":"Power-saving bounds for Thue--Mahler and Mordell equations","publication_year":2026,"publication_date":"2026-08-24","ids":{"openalex":"https://openalex.org/W7204171943","doi":"https://doi.org/10.48550/arxiv.2608.23559"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.23559","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.23559","pdf_url":"https://arxiv.org/pdf/2608.23559","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.23559","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012806097","display_name":"H\u00e9ctor Past\u00e9n","orcid":"https://orcid.org/0000-0002-7789-8459","observed_orcids":["https://orcid.org/0000-0002-7789-8459"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Pasten, Hector","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5012806097"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9199000000953674,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9199000000953674,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.07620000094175339,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.0010000000474974513,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mordell-equation","display_name":"Mordell equation","score":0.9319999814033508},{"id":"https://openalex.org/keywords/regulators","display_name":"regulators","score":0.6269999742507935}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7730000019073486},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6635000109672546},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6498000025749207},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5634999871253967},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5332000255584717},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.4007999897003174},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3725000023841858},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3625999987125397},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.36250001192092896},{"id":"https://openalex.org/C34800285","wikidata":"https://www.wikidata.org/wiki/Q5159395","display_name":"Conductor","level":2,"score":0.35690000653266907},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.3377000093460083},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.3276999890804291},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.32760000228881836},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.3100999891757965},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28949999809265137},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.2818000018596649},{"id":"https://openalex.org/C82558694","wikidata":"https://www.wikidata.org/wiki/Q1545619","display_name":"Aspect ratio (aeronautics)","level":2,"score":0.2766999900341034}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.23559","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.23559","pdf_url":"https://arxiv.org/pdf/2608.23559","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.23559","doi":"https://doi.org/10.48550/arxiv.2608.23559","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.23559","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.23559","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.23559","pdf_url":"https://arxiv.org/pdf/2608.23559","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1767060656","display_name":"DIOPHANTINE PROBLEMS ON RATIONAL AND INTEGRAL POINTS","funder_award_id":"1230507","funder_id":"https://openalex.org/F4320331146","funder_display_name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo"}],"funders":[{"id":"https://openalex.org/F4320331146","display_name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","ror":"https://ror.org/05btm5233"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204171943.pdf","grobid_xml":"https://content.openalex.org/works/W7204171943.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2519356009","https://openalex.org/W3102327505","https://openalex.org/W1527292340","https://openalex.org/W6948359777","https://openalex.org/W2050599248","https://openalex.org/W6891719869","https://openalex.org/W2789322254","https://openalex.org/W2949579080","https://openalex.org/W2997148872","https://openalex.org/W7213391143"],"abstract_inverted_index":{"We":[0],"prove":[1],"a":[2,37,93],"new":[3],"effective":[4],"bound":[5,24,79],"for":[6,28,63,106],"cubic":[7],"Thue--Mahler":[8],"equations":[9],"with":[10,123],"power-saving":[11,51,94],"dependence":[12],"on":[13,55],"the":[14,29,42,49,74,97,101,116],"regulator.":[15],"This":[16],"has":[17],"some":[18],"applications.":[19],"First,":[20],"we":[21,72,91],"improve":[22,73],"Stark's":[23],"$\\log":[25],"\\max\\{|x|,|y|\\}":[26],"\\ll_\u03b5|k|^{1+\u03b5}$":[27],"integer":[30,64],"solutions":[31],"of":[32,68,100,112,118],"Mordell's":[33],"equation":[34],"$y^2=x^3+k$":[35],"($k$":[36],"non-zero":[38],"integer)":[39],"by":[40,83],"reducing":[41],"exponent":[43],"$1+\u03b5$":[44],"to":[45,87],"$1/2+\u03b5$;":[46],"this":[47],"is":[48],"first":[50],"improvement":[52,95],"without":[53],"restrictions":[54],"$k$":[56],"in":[57,85,96,115],"more":[58],"than":[59],"50":[60],"years.":[61],"Secondly,":[62],"squares":[65],"and":[66],"cubes":[67],"size":[69],"$\\asymp":[70],"T$":[71],"known":[75],"unconditional":[76],"separation":[77],"lower":[78],"$(\\log":[80,88],"T)^{1-o(1)}$":[81],"obtained":[82],"Stark":[84],"1973":[86],"T)^{2-o(1)}$.":[89],"Finally,":[90],"obtain":[92],"conductor":[98],"aspect":[99],"strongest":[102],"currently":[103],"available":[104],"bounds":[105],"Frey's":[107],"height":[108],"conjecture":[109],"(a":[110],"strengthening":[111],"Szpiro's":[113],"conjecture)":[114],"case":[117],"elliptic":[119],"curves":[120],"over":[121],"$\\mathbb{Q}$":[122],"integral":[124],"$j$-invariant.":[125]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-26T00:00:00"}