{"id":"https://openalex.org/W7214781791","doi":"https://doi.org/10.48550/arxiv.2609.34551","title":"Interlacing on the Unit Circle via Coefficientwise Reciprocals","display_name":"Interlacing on the Unit Circle via Coefficientwise Reciprocals","publication_year":2026,"publication_date":"2026-09-28","ids":{"openalex":"https://openalex.org/W7214781791","doi":"https://doi.org/10.48550/arxiv.2609.34551"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.34551","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.34551","pdf_url":"https://arxiv.org/pdf/2609.34551","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/2609.34551","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5063617101","display_name":"Jianxi Mao","orcid":"https://orcid.org/0000-0003-0673-4964","observed_orcids":["https://orcid.org/0000-0003-0673-4964"]},"institutions":[{"id":"https://openalex.org/I27357992","display_name":"Dalian University of Technology","ror":"https://ror.org/023hj5876","country_code":"CN","type":"education","lineage":["https://openalex.org/I27357992"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Mao, Jianxi","raw_affiliation_strings":["School of Mathematical Sciences , Dalian University of Technology , Dalian 116024 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Dalian University of Technology , Dalian 116024 , P. R. China","institution_ids":["https://openalex.org/I27357992"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5140010735","display_name":"Lijie Wang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I27357992","display_name":"Dalian University of Technology","ror":"https://ror.org/023hj5876","country_code":"CN","type":"education","lineage":["https://openalex.org/I27357992"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Lijie","raw_affiliation_strings":["School of Mathematical Sciences , Dalian University of Technology , Dalian 116024 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Dalian University of Technology , Dalian 116024 , P. R. China","institution_ids":["https://openalex.org/I27357992"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5153534090","display_name":"Sainan Zheng","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I130480701","display_name":"Dongbei University of Finance and Economics","ror":"https://ror.org/05db1pj03","country_code":"CN","type":"education","lineage":["https://openalex.org/I130480701"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zheng, Sainan","raw_affiliation_strings":["School of Data Science and Artificial Intelligence , Dongbei University of Finance and Economics , Dalian 116025 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Data Science and Artificial Intelligence , Dongbei University of Finance and Economics , Dalian 116025 , P. R. China","institution_ids":["https://openalex.org/I130480701"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.5640000104904175,"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":0.5640000104904175,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.396699994802475,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.007300000172108412,"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/zeros-on-the-unit-circle","display_name":"unit circle zeros","score":0.9160000085830688},{"id":"https://openalex.org/keywords/palindromic-polynomials","display_name":"palindromic polynomials","score":0.9340000152587891},{"id":"https://openalex.org/keywords/finite-blaschke-products","display_name":"finite Blaschke products","score":0.9639999866485596},{"id":"https://openalex.org/keywords/boundary-phase","display_name":"boundary phase","score":0.7710000276565552},{"id":"https://openalex.org/keywords/rogers-szego-polynomials","display_name":"Rogers-Szeg\u0151 polynomials","score":0.9190000295639038},{"id":"https://openalex.org/keywords/simple-zeros","display_name":"simple zeros","score":0.640999972820282}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8722000122070312},{"id":"https://openalex.org/C95439218","wikidata":"https://www.wikidata.org/wiki/Q1666173","display_name":"Interlacing","level":2,"score":0.8094000220298767},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.7975999712944031},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.7904000282287598},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.586899995803833},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5777000188827515},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5349000096321106},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.4832000136375427},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4652999937534332},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4002000093460083},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.37220001220703125},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.3474000096321106},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.3449000120162964},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3337000012397766},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.29660001397132874},{"id":"https://openalex.org/C44667518","wikidata":"https://www.wikidata.org/wiki/Q12321","display_name":"Palindrome","level":4,"score":0.29249998927116394},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.2921999990940094},{"id":"https://openalex.org/C47177299","wikidata":"https://www.wikidata.org/wiki/Q2390323","display_name":"Unit disk","level":2,"score":0.2912999987602234},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2549999952316284}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.34551","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.34551","pdf_url":"https://arxiv.org/pdf/2609.34551","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.2609.34551","doi":"https://doi.org/10.48550/arxiv.2609.34551","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.34551","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.34551","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.34551","pdf_url":"https://arxiv.org/pdf/2609.34551","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/W7214781791.pdf","grobid_xml":"https://content.openalex.org/works/W7214781791.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2968546255","https://openalex.org/W7128474359"],"related_works":["https://openalex.org/W2004287231","https://openalex.org/W2314485566","https://openalex.org/W2048520643","https://openalex.org/W106033339","https://openalex.org/W2054591065","https://openalex.org/W2963943352","https://openalex.org/W1988444350","https://openalex.org/W2061476147","https://openalex.org/W4410328909","https://openalex.org/W205055400"],"abstract_inverted_index":{"Let":[0],"$f(z)=\\sum_{k=0}^{n}a_kz^k$":[1],"be":[2],"a":[3,112],"polynomial":[4],"with":[5,55,118],"positive":[6],"coefficients,":[7],"and":[8,23,44,52,68,85,106,130],"define":[9],"its":[10],"coefficientwise":[11,62,119],"reciprocal":[12,102,104,107],"by":[13],"$f^{\\#}(z)=\\sum_{k=0}^{n}\\frac{z^k}{a_k}.$":[14],"It":[15],"is":[16,21,79],"known":[17],"that":[18,41],"if":[19,42],"$f$":[20],"palindromic":[22,47],"has":[24],"only":[25,56,65],"negative":[26,57],"real":[27,58],"zeros,":[28,59,67],"then":[29,60],"all":[30],"zeros":[31],"of":[32,49,88],"$f^{\\#}$":[33],"lie":[34],"on":[35,73,81],"the":[36,74,86,101,116],"unit":[37,75],"circle.":[38,76],"We":[39],"prove":[40],"$p$":[43],"$q$":[45],"are":[46],"polynomials":[48],"degrees":[50],"$n$":[51],"$n+1$,":[53],"respectively,":[54],"their":[61,89],"reciprocals":[63],"have":[64],"simple":[66],"$p^{\\#}$":[69],"strictly":[70,124],"interlaces":[71],"$q^{\\#}$":[72],"Our":[77],"proof":[78],"based":[80],"finite":[82],"Blaschke":[83],"products":[84],"comparison":[87],"boundary":[90],"phases.":[91],"As":[92],"an":[93],"immediate":[94],"consequence,":[95],"we":[96,121],"obtain":[97],"strict":[98],"interlacing":[99,125],"for":[100],"binomial,":[103],"Eulerian,":[105],"Narayana":[108],"polynomials.":[109,132],"By":[110],"combining":[111],"sign":[113],"change":[114],"in":[115],"$\u03b3$-coefficients":[117],"reciprocation,":[120],"further":[122],"construct":[123],"families":[126],"from":[127],"Rogers--Szeg\u0151,":[128],"Poupard,":[129],"Kreweras-related":[131]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}
