{"id":"https://openalex.org/W7207878281","doi":"https://doi.org/10.48550/arxiv.2609.02871","title":"Estimating the number of real zeros of linear combinations of radicals of polynomials","display_name":"Estimating the number of real zeros of linear combinations of radicals of polynomials","publication_year":2026,"publication_date":"2026-09-02","ids":{"openalex":"https://openalex.org/W7207878281","doi":"https://doi.org/10.48550/arxiv.2609.02871"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.02871","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02871","pdf_url":"https://arxiv.org/pdf/2609.02871","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2609.02871","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5055754292","display_name":"Gal Binyamini","orcid":"https://orcid.org/0000-0002-2714-4106","observed_orcids":["https://orcid.org/0000-0002-2714-4106"]},"institutions":[{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Binyamini, Gal","raw_affiliation_strings":["Department of Mathematics , Weizmann Institute of Science , Rehovot , Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Weizmann Institute of Science , Rehovot , Israel","institution_ids":["https://openalex.org/I53964585"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056976182","display_name":"Avner Kiro","orcid":"https://orcid.org/0000-0002-5072-8089","observed_orcids":["https://orcid.org/0000-0002-5072-8089"]},"institutions":[{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Kiro, Avner","raw_affiliation_strings":["Department of Mathematics , Weizmann Institute of Science , Rehovot , Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Weizmann Institute of Science , Rehovot , Israel","institution_ids":["https://openalex.org/I53964585"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024942308","display_name":"Alexander Logunov","orcid":"https://orcid.org/0000-0002-9178-2671","observed_orcids":["https://orcid.org/0000-0002-9178-2671"]},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Logunov, Alexander","raw_affiliation_strings":["Department of Mathematics , Massachusetts Institute of Technology , Cambridge , MA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Massachusetts Institute of Technology , Cambridge , MA , USA","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102749597","display_name":"Dmitry Novikov","orcid":"https://orcid.org/0000-0002-9182-5344","observed_orcids":["https://orcid.org/0000-0002-9182-5344"]},"institutions":[{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Novikov, Dmitry","raw_affiliation_strings":["Department of Mathematics , Weizmann Institute of Science , Rehovot , Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Weizmann Institute of Science , Rehovot , Israel","institution_ids":["https://openalex.org/I53964585"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010422308","display_name":"Dmitrii Zakharov","orcid":"https://orcid.org/0000-0001-9594-9144","observed_orcids":["https://orcid.org/0000-0001-9594-9144","https://orcid.org/0009-0007-6369-7190"]},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zakharov, Dmitrii","raw_affiliation_strings":["Department of Mathematics , Massachusetts Institute of Technology , Cambridge , MA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Massachusetts Institute of Technology , Cambridge , MA , USA","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":2,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.3107999861240387,"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"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.3107999861240387,"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"}},{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.2863999903202057,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.13500000536441803,"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/real-zeros-of-polynomials","display_name":"real zeros of polynomials","score":0.7540000081062317},{"id":"https://openalex.org/keywords/gaussian-mixture-model","display_name":"Gaussian mixture model","score":0.6359999775886536},{"id":"https://openalex.org/keywords/critical-points","display_name":"critical points","score":0.8999999761581421},{"id":"https://openalex.org/keywords/nodal-domains","display_name":"nodal domains","score":0.8659999966621399},{"id":"https://openalex.org/keywords/laplace-beltrami-eigenfunctions","display_name":"Laplace-Beltrami eigenfunctions","score":0.9490000009536743}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.888700008392334},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.541700005531311},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5321999788284302},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.49959999322891235},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.44290000200271606},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4250999987125397},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4237000048160553},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.40950000286102295},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.38690000772476196},{"id":"https://openalex.org/C17103678","wikidata":"https://www.wikidata.org/wiki/Q1474074","display_name":"Homogeneous polynomial","level":4,"score":0.3855000138282776},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.36660000681877136},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.36550000309944153},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.3610000014305115},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.33880001306533813},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3336000144481659},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33079999685287476},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.3296999931335449},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.29919999837875366},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.2978000044822693},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.2976999878883362},{"id":"https://openalex.org/C150397156","wikidata":"https://www.wikidata.org/wiki/Q591310","display_name":"General position","level":2,"score":0.28600001335144043},{"id":"https://openalex.org/C163635466","wikidata":"https://www.wikidata.org/wiki/Q3099696","display_name":"Monic polynomial","level":3,"score":0.27379998564720154},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.2718000113964081},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.2703999876976013},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.26089999079704285},{"id":"https://openalex.org/C139505817","wikidata":"https://www.wikidata.org/wiki/Q5527834","display_name":"Gaussian binomial coefficient","level":4,"score":0.25940001010894775},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.257099986076355},{"id":"https://openalex.org/C20050597","wikidata":"https://www.wikidata.org/wiki/Q261527","display_name":"Linear form","level":2,"score":0.2551000118255615}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.02871","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02871","pdf_url":"https://arxiv.org/pdf/2609.02871","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.02871","doi":"https://doi.org/10.48550/arxiv.2609.02871","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.02871","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.02871","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02871","pdf_url":"https://arxiv.org/pdf/2609.02871","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320322596","display_name":"Tel Aviv University","ror":"https://ror.org/04mhzgx49"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7207878281.pdf","grobid_xml":"https://content.openalex.org/works/W7207878281.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2605641194","https://openalex.org/W7172068017"],"related_works":["https://openalex.org/W1993146230","https://openalex.org/W2039902468","https://openalex.org/W2705674073","https://openalex.org/W2072184892","https://openalex.org/W2436246852","https://openalex.org/W2978031719","https://openalex.org/W7128144941","https://openalex.org/W2416112264","https://openalex.org/W7099168775","https://openalex.org/W89656303"],"abstract_inverted_index":{"We":[0,47],"obtain":[1],"upper":[2,52],"bounds":[3,53,62],"for":[4,54,109,141,156],"the":[5,13,55,74,91,103,132,142,157,165,168,182,190,209],"number":[6,56,143,169],"of":[7,10,12,34,57,73,80,83,98,102,112,144,147,159,170,173,189],"real":[8,32],"zeros":[9],"functions":[11,202],"form":[14],"$$":[15,21],"f(x)":[16],"=":[17,179],"\\sum_{k=1}^{n}":[18],"c_k":[19],"\\bigl(P_k(x)\\bigr)^{\u03b1_k},":[20],"where":[22],"$c_k,":[23],"\u03b1_k":[24],"\\in":[25],"\\mathbb{R}$":[26],"and":[27,71,115,152,205],"each":[28],"$P_k$":[29],"is":[30,40],"a":[31,96,106,110,138,153,187,195],"polynomial":[33,65,139],"degree":[35],"at":[36],"most":[37],"$d$":[38],"that":[39,63],"non-negative":[41],"on":[42,59,87,181,203,208],"an":[43,127],"interval":[44],"$I\\subset":[45],"\\mathbb{R}$.":[46],"improve":[48],"previously":[49],"known":[50],"exponential":[51],"roots":[58,82],"$I$":[60],"to":[61,118,167,175],"are":[64],"in":[66,69,131],"$n$,":[67],"linear":[68,78,92,107],"$d$,":[70],"independent":[72,125],"exponents":[75],"$\u03b1_k$.":[76],"For":[77],"combinations":[79],"square":[81],"positive":[84,160],"quadratic":[85,161],"polynomials":[86],"$\\mathbb{R}$":[88],"we":[89,193],"prove":[90],"bound":[93,108,140],"$2n$,":[94],"answering":[95],"question":[97,111],"N.~Alon.":[99],"A":[100],"modification":[101],"argument":[104],"yields":[105],"A.~Gabrielov,":[113],"D.~Novikov,":[114],"B.~Shapiro":[116],"related":[117],"Maxwell's":[119],"conjecture.":[120],"The":[121],"article":[122],"describes":[123],"two":[124],"approaches:":[126],"elementary":[128],"ODE":[129],"method":[130,155],"general":[133],"case,":[134],"which":[135,163],"also":[136],"gives":[137],"critical":[145],"points":[146],"one":[148],"dimensional":[149],"Gaussian":[150],"mixtures,":[151],"PDE":[154],"case":[158],"polynomials,":[162],"connects":[164],"problem":[166],"nodal":[171],"domains":[172],"solutions":[174],"$\u0394u":[176],"+":[177],"\u03bbu":[178],"0$":[180],"punctured":[183],"hyperbolic":[184,210],"plane.":[185],"As":[186],"byproduct":[188],"second":[191],"approach,":[192],"describe":[194],"curious":[196],"relation":[197],"between":[198],"axially":[199],"symmetric":[200],"harmonic":[201],"$\\mathbb{R}^3\\setminus\\{(x,0,0)\\}$":[204],"Laplace-Beltrami":[206],"eigenfunctions":[207],"plane":[211],"with":[212],"eigenvalue":[213],"$1/4$.":[214]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-04T00:00:00"}