{"id":"https://openalex.org/W7214918501","doi":"https://doi.org/10.48550/arxiv.2609.37180","title":"Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc","display_name":"Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc","publication_year":2026,"publication_date":"2026-09-29","ids":{"openalex":"https://openalex.org/W7214918501","doi":"https://doi.org/10.48550/arxiv.2609.37180"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.37180","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.37180","pdf_url":"https://arxiv.org/pdf/2609.37180","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/2609.37180","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153655357","display_name":"Oliver Nav\u00edo","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nav\u00edo, Oliver","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5048730095","display_name":"Daniel Seco","orcid":"https://orcid.org/0000-0002-1634-2318","observed_orcids":["https://orcid.org/0000-0002-1634-2318"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Seco, Daniel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/T10884","display_name":"Holomorphic and Operator Theory","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10884","display_name":"Holomorphic and Operator Theory","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11191","display_name":"Mathematical functions and polynomials","score":0.0010999999940395355,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11822","display_name":"Analytic and geometric function theory","score":0.0005000000237487257,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/dirichlet-type-spaces","display_name":"Dirichlet-type spaces","score":0.9589999914169312},{"id":"https://openalex.org/keywords/bidisc","display_name":"bidisc","score":0.9359999895095825},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.7480000257492065},{"id":"https://openalex.org/keywords/reciprocal-functions","display_name":"reciprocal functions","score":0.41999998688697815},{"id":"https://openalex.org/keywords/symmetric-polynomials","display_name":"symmetric polynomials","score":0.6949999928474426},{"id":"https://openalex.org/keywords/polydisk","display_name":"polydisk","score":0.7009999752044678},{"id":"https://openalex.org/keywords/reproducing-kernel-hilbert-space","display_name":"reproducing kernel Hilbert space","score":0.574999988079071}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8996000289916992},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.7745000123977661},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6608999967575073},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.569599986076355},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.524399995803833},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.5149000287055969},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.5113000273704529},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.4869000017642975},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.48539999127388},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.47130000591278076},{"id":"https://openalex.org/C17103678","wikidata":"https://www.wikidata.org/wiki/Q1474074","display_name":"Homogeneous polynomial","level":4,"score":0.4327999949455261},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.428600013256073},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.398499995470047},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.38040000200271606},{"id":"https://openalex.org/C97395012","wikidata":"https://www.wikidata.org/wiki/Q1422356","display_name":"Reciprocal polynomial","level":4,"score":0.3797000050544739},{"id":"https://openalex.org/C170412648","wikidata":"https://www.wikidata.org/wiki/Q4736415","display_name":"Alternating polynomial","level":4,"score":0.3776000142097473},{"id":"https://openalex.org/C30597210","wikidata":"https://www.wikidata.org/wiki/Q1163608","display_name":"Minimal polynomial (linear algebra)","level":5,"score":0.365200012922287},{"id":"https://openalex.org/C163635466","wikidata":"https://www.wikidata.org/wiki/Q3099696","display_name":"Monic polynomial","level":3,"score":0.3596999943256378},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.34599998593330383},{"id":"https://openalex.org/C176370821","wikidata":"https://www.wikidata.org/wiki/Q1826459","display_name":"Linear space","level":2,"score":0.33970001339912415},{"id":"https://openalex.org/C45025165","wikidata":"https://www.wikidata.org/wiki/Q13424738","display_name":"Stable polynomial","level":5,"score":0.33180001378059387},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.32190001010894775},{"id":"https://openalex.org/C42321380","wikidata":"https://www.wikidata.org/wiki/Q849705","display_name":"Characteristic polynomial","level":3,"score":0.3208000063896179},{"id":"https://openalex.org/C173110770","wikidata":"https://www.wikidata.org/wiki/Q382698","display_name":"Affine space","level":3,"score":0.3034000098705292},{"id":"https://openalex.org/C101525915","wikidata":"https://www.wikidata.org/wiki/Q7581979","display_name":"Square-free polynomial","level":4,"score":0.2784999907016754},{"id":"https://openalex.org/C20050597","wikidata":"https://www.wikidata.org/wiki/Q261527","display_name":"Linear form","level":2,"score":0.2712000012397766},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.2612000107765198},{"id":"https://openalex.org/C52153879","wikidata":"https://www.wikidata.org/wiki/Q930499","display_name":"Symmetric polynomial","level":4,"score":0.2513999938964844}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.37180","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.37180","pdf_url":"https://arxiv.org/pdf/2609.37180","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.2609.37180","doi":"https://doi.org/10.48550/arxiv.2609.37180","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.37180","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:2609.37180","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.37180","pdf_url":"https://arxiv.org/pdf/2609.37180","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214918501.pdf","grobid_xml":"https://content.openalex.org/works/W7214918501.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4409729980","https://openalex.org/W4401339102","https://openalex.org/W4289294801","https://openalex.org/W2465162145","https://openalex.org/W1564066951","https://openalex.org/W2220959729","https://openalex.org/W4223652338","https://openalex.org/W1506407083","https://openalex.org/W3156899961","https://openalex.org/W4416363418"],"abstract_inverted_index":{"We":[0,45],"study":[1],"optimal":[2,19,73],"polynomial":[3,20,29,60,83],"approximants":[4],"in":[5,41],"spaces":[6],"of":[7,52,88],"Dirichlet-type":[8,54,106],"over":[9],"the":[10,17,27,37,42,53,66,70,79,86,89],"bidisc.":[11,80],"Given":[12],"a":[13,58,76],"nonzero":[14],"function":[15,39],"f,":[16],"linear":[18,72],"approximant":[21,74],"(opa)":[22],"to":[23,36,85,100,104],"its":[24],"reciprocal":[25],"is":[26,34],"affine":[28],"p":[30],"for":[31,48],"which":[32],"pf":[33],"closest":[35],"constant":[38],"1":[40],"space":[43],"norm.":[44],"prove":[46],"that,":[47],"every":[49],"positive":[50],"value":[51],"parameter,":[55],"there":[56],"exists":[57],"symmetric":[59],"f":[61],"with":[62],"no":[63],"zeros":[64],"on":[65,93],"closed":[67],"bidisc,":[68],"while":[69],"corresponding":[71],"has":[75],"zero":[77],"inside":[78],"This":[81],"provides":[82],"counterexamples":[84],"version":[87],"Weak":[90],"Shanks":[91],"Conjecture":[92],"these":[94],"spaces.":[95,107],"The":[96],"construction":[97],"also":[98],"extends":[99],"higher-dimensional":[101],"polydiscs":[102],"and":[103],"anisotropic":[105]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-01T00:00:00"}
