{"id":"https://openalex.org/W2079539759","doi":"https://doi.org/10.1112/jlms/jdq061","title":"Algebro-geometric aspects of Heine-Stieltjes theory","display_name":"Algebro-geometric aspects of Heine-Stieltjes theory","publication_year":2010,"publication_date":"2010-11-23","ids":{"openalex":"https://openalex.org/W2079539759","doi":"https://doi.org/10.1112/jlms/jdq061","mag":"2079539759"},"language":"en","primary_location":{"id":"doi:10.1112/jlms/jdq061","is_oa":false,"landing_page_url":"https://doi.org/10.1112/jlms/jdq061","pdf_url":null,"source":{"id":"https://openalex.org/S4210207695","display_name":"Journal of the London Mathematical Society","issn_l":"0024-6107","issn":["0024-6107","1469-7750"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of the London Mathematical Society","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/0812.4193","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041917385","display_name":"Boris Shapiro","orcid":"https://orcid.org/0000-0002-8438-3971"},"institutions":[{"id":"https://openalex.org/I161593684","display_name":"Stockholm University","ror":"https://ror.org/05f0yaq80","country_code":"SE","type":"education","lineage":["https://openalex.org/I161593684"]}],"countries":["SE"],"is_corresponding":true,"raw_author_name":"Boris Shapiro","raw_affiliation_strings":["Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden","Department of Mathematics Stockholm University SE-106 91 Stockholm Sweden"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden","institution_ids":["https://openalex.org/I161593684"]},{"raw_affiliation_string":"Department of Mathematics Stockholm University SE-106 91 Stockholm Sweden","institution_ids":["https://openalex.org/I161593684"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5041917385"],"corresponding_institution_ids":["https://openalex.org/I161593684"],"apc_list":{"value":3710,"currency":"USD","value_usd":3710},"apc_paid":null,"fwci":4.2457,"has_fulltext":false,"cited_by_count":39,"citation_normalized_percentile":{"value":0.94278903,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"83","issue":"1","first_page":"36","last_page":"56"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9990000128746033,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9990000128746033,"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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9825000166893005,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.7513911724090576},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7343536615371704},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.592251181602478},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5287266373634338},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5269349813461304},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5243204832077026},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.47686490416526794},{"id":"https://openalex.org/keywords/degeneracy","display_name":"Degeneracy (biology)","score":0.47222283482551575},{"id":"https://openalex.org/keywords/riemann\u2013stieltjes-integral","display_name":"Riemann\u2013Stieltjes integral","score":0.4478621184825897},{"id":"https://openalex.org/keywords/univariate","display_name":"Univariate","score":0.4211379289627075},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40271347761154175},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3243524432182312},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23932260274887085},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13284489512443542},{"id":"https://openalex.org/keywords/multivariate-statistics","display_name":"Multivariate statistics","score":0.06763046979904175},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.06607285141944885}],"concepts":[{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.7513911724090576},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7343536615371704},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.592251181602478},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5287266373634338},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5269349813461304},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5243204832077026},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.47686490416526794},{"id":"https://openalex.org/C2777727622","wikidata":"https://www.wikidata.org/wiki/Q5251772","display_name":"Degeneracy (biology)","level":2,"score":0.47222283482551575},{"id":"https://openalex.org/C167211080","wikidata":"https://www.wikidata.org/wiki/Q285771","display_name":"Riemann\u2013Stieltjes integral","level":3,"score":0.4478621184825897},{"id":"https://openalex.org/C199163554","wikidata":"https://www.wikidata.org/wiki/Q1681619","display_name":"Univariate","level":3,"score":0.4211379289627075},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40271347761154175},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3243524432182312},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23932260274887085},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13284489512443542},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.06763046979904175},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.06607285141944885},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C60644358","wikidata":"https://www.wikidata.org/wiki/Q128570","display_name":"Bioinformatics","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1112/jlms/jdq061","is_oa":false,"landing_page_url":"https://doi.org/10.1112/jlms/jdq061","pdf_url":null,"source":{"id":"https://openalex.org/S4210207695","display_name":"Journal of the London Mathematical Society","issn_l":"0024-6107","issn":["0024-6107","1469-7750"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of the London Mathematical Society","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0812.4193","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0812.4193","pdf_url":"https://arxiv.org/pdf/0812.4193","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,"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":"pmh:oai:CiteSeerX.psu:10.1.1.145.4340","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.145.4340","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.matematik.su.se/~shapiro/Articles/AlgAspectsFin.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.250.2756","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.250.2756","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0812.4193v1.pdf","raw_type":"text"},{"id":"pmh:oai:DiVA.org:su-49772","is_oa":true,"landing_page_url":"http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-49772","pdf_url":null,"source":{"id":"https://openalex.org/S4306400561","display_name":"DiVA (Stockholm University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I161593684","host_organization_name":"Stockholm University","host_organization_lineage":["https://openalex.org/I161593684"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:0812.4193","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0812.4193","pdf_url":"https://arxiv.org/pdf/0812.4193","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,"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":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":52,"referenced_works":["https://openalex.org/W27641535","https://openalex.org/W45833573","https://openalex.org/W85287489","https://openalex.org/W108024965","https://openalex.org/W140650899","https://openalex.org/W565222581","https://openalex.org/W625604597","https://openalex.org/W1502260107","https://openalex.org/W1585998846","https://openalex.org/W1603002419","https://openalex.org/W1777425630","https://openalex.org/W1968111761","https://openalex.org/W1969875028","https://openalex.org/W1970962195","https://openalex.org/W1976182206","https://openalex.org/W1977012058","https://openalex.org/W1977415556","https://openalex.org/W1991711762","https://openalex.org/W2000628644","https://openalex.org/W2012187381","https://openalex.org/W2014075178","https://openalex.org/W2014114274","https://openalex.org/W2017699071","https://openalex.org/W2022076748","https://openalex.org/W2024358108","https://openalex.org/W2025299391","https://openalex.org/W2029306691","https://openalex.org/W2039343706","https://openalex.org/W2040573339","https://openalex.org/W2047850430","https://openalex.org/W2050033941","https://openalex.org/W2055760212","https://openalex.org/W2073020812","https://openalex.org/W2135009399","https://openalex.org/W2142974299","https://openalex.org/W2159148117","https://openalex.org/W2161543402","https://openalex.org/W2334347857","https://openalex.org/W2949754518","https://openalex.org/W2952201608","https://openalex.org/W2964020078","https://openalex.org/W2964332565","https://openalex.org/W3102020300","https://openalex.org/W3122550644","https://openalex.org/W3144205924","https://openalex.org/W4234139065","https://openalex.org/W4237125420","https://openalex.org/W4240867252","https://openalex.org/W4243146230","https://openalex.org/W4244191199","https://openalex.org/W4250858877","https://openalex.org/W4394644196"],"related_works":["https://openalex.org/W2555414267","https://openalex.org/W52186027","https://openalex.org/W1828158523","https://openalex.org/W2372940580","https://openalex.org/W4224324299","https://openalex.org/W2047547195","https://openalex.org/W204175656","https://openalex.org/W4386761228","https://openalex.org/W2016679969","https://openalex.org/W2053002746"],"abstract_inverted_index":{"The":[0],"goal":[1],"of":[2,16,83,128,133,172,202,209],"this":[3,213],"paper":[4],"was":[5],"to":[6],"develop":[7],"a":[8,20,74,141,168,207],"Heine\u2013Stieltjes":[9],"theory":[10],"for":[11,140],"univariate":[12],"linear":[13,22],"differential":[14,24],"operators":[15],"higher":[17,76],"order.":[18],"For":[19],"given":[21,142],"ordinary":[23],"operator":[25],"\ud835\udd21":[26,148],"(":[27,38,41,149,153,158,162,186],"z":[28,39,46,150,154,159,163],")":[29,40,48,151,155,160,164,191],"=":[30,33,165],"\u2211":[31],"i":[32,37,43,47],"1":[34,88],"k":[35,57],"Q":[36],"d":[42,45,49],"/":[44],"with":[50],"polynomial":[51,169],"coefficients,":[52],"set":[53],"r=max":[54],"i=1,":[55],"\u2026,":[56],"(deg":[58],"Qi(z)\u2212i).":[59],"If":[60],"\ud835\udd21(z)":[61],"satisfies":[62],"conditions":[63],"(i)":[64],"r\u2a7e0":[65],"and":[66,94,215],"(ii)":[67],"deg":[68],"Qk(z)=k+r,":[69],"then":[70],"we":[71,122],"call":[72],"it":[73],"non-degenerate":[75],"Lam\u00e9":[77],"operator.":[78],"Following":[79],"the":[80,124,146],"classical":[81],"approach":[82],"Heine":[84],"[Handbuch":[85],"der":[86],"Kugelfunctionen":[87],"(G.":[89],"Reimer,":[90],"Berlin,":[91],"1878)":[92],"472\u2013479]":[93],"Stieltjes":[95],"[\u2018Sur":[96],"certains":[97],"polyn\u00f4mes":[98],"qui":[99],"v\u00e9rifient":[100],"une":[101],"\u00e9quation":[102],"diff\u00e9rentielle":[103],"lin\u00e9aire":[104],"du":[105],"second":[106],"ordre":[107],"et":[108],"sur":[109],"la":[110],"theorie":[111],"des":[112],"fonctions":[113],"de":[114],"Lam\u00e9\u2019,":[115],"Acta":[116],"Math.":[117],"6":[118],"(1885)":[119],"321\u2013326":[120],"(French)],":[121],"study":[123],"multiparameter":[125],"spectral":[126],"problem":[127],"finding":[129],"all":[130],"polynomials":[131,193],"V(z)":[132],"degree":[134,173,203],"at":[135],"most":[136],"r":[137,190],"such":[138,192],"that":[139,177],"positive":[143],"integer":[144],"n,":[145],"equation":[147],"S":[152,161],"+":[156,189],"V":[157],"0":[166],"has":[167],"solution":[170],"S(z)":[171],"n.":[174,204],"We":[175,205],"show":[176],"under":[178],"some":[179],"mild":[180],"non-degeneracy":[181],"assumptions,":[182],"there":[183],"exist":[184],"exactly":[185],"n":[187,188],"Vn,":[194],"i(z)":[195,200],"whose":[196],"corresponding":[197],"eigenpolynomials":[198],"Sn,":[199],"are":[201],"generalize":[206],"number":[208],"well-known":[210],"results":[211],"in":[212],"area":[214],"discuss":[216],"occurring":[217],"degeneracies.":[218]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":2}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2025-10-10T00:00:00"}