{"id":"https://openalex.org/W7214290495","doi":"https://doi.org/10.48550/arxiv.2609.29655","title":"Beyond Harnack Rigidity","display_name":"Beyond Harnack Rigidity","publication_year":2026,"publication_date":"2026-08-29","ids":{"openalex":"https://openalex.org/W7214290495","doi":"https://doi.org/10.48550/arxiv.2609.29655"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.29655","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29655","pdf_url":"https://arxiv.org/pdf/2609.29655","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/2609.29655","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013137097","display_name":"Mounir Nisse","orcid":"https://orcid.org/0000-0003-0455-0600","observed_orcids":["https://orcid.org/0000-0003-0455-0600"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Nisse, Mounir","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5013137097"],"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":0.9810000061988831,"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":0.9810000061988831,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.014000000432133675,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.0008999999845400453,"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/harnack-curves","display_name":"Harnack curves","score":0.9559999704360962},{"id":"https://openalex.org/keywords/lattice-polygons","display_name":"lattice polygons","score":0.9120000004768372},{"id":"https://openalex.org/keywords/real-plane-algebraic-curves","display_name":"real plane algebraic curves","score":0.6029999852180481}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7350999712944031},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.5875999927520752},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5534999966621399},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.436599999666214},{"id":"https://openalex.org/C156004811","wikidata":"https://www.wikidata.org/wiki/Q2228257","display_name":"Multiplicity (mathematics)","level":2,"score":0.4207000136375427},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.40119999647140503},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4004000127315521},{"id":"https://openalex.org/C58315980","wikidata":"https://www.wikidata.org/wiki/Q179436","display_name":"Asymptote","level":2,"score":0.38449999690055847},{"id":"https://openalex.org/C2776809875","wikidata":"https://www.wikidata.org/wiki/Q1375963","display_name":"Converse","level":2,"score":0.37290000915527344},{"id":"https://openalex.org/C8464174","wikidata":"https://www.wikidata.org/wiki/Q868279","display_name":"Harnack's inequality","level":2,"score":0.33980000019073486},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.3206000030040741},{"id":"https://openalex.org/C43809302","wikidata":"https://www.wikidata.org/wiki/Q1060874","display_name":"Plane curve","level":2,"score":0.3163999915122986},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.30709999799728394},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.303600013256073},{"id":"https://openalex.org/C70251129","wikidata":"https://www.wikidata.org/wiki/Q260928","display_name":"Jordan curve theorem","level":2,"score":0.30079999566078186},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.298799991607666},{"id":"https://openalex.org/C167204820","wikidata":"https://www.wikidata.org/wiki/Q912058","display_name":"Simply connected space","level":2,"score":0.296999990940094},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.28529998660087585},{"id":"https://openalex.org/C163072092","wikidata":"https://www.wikidata.org/wiki/Q1135706","display_name":"Harnack's principle","level":3,"score":0.2849000096321106},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.2797999978065491},{"id":"https://openalex.org/C84597430","wikidata":"https://www.wikidata.org/wiki/Q106227","display_name":"Locus (genetics)","level":3,"score":0.2689000070095062},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.26330000162124634},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.25600001215934753},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.2524000108242035}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.29655","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29655","pdf_url":"https://arxiv.org/pdf/2609.29655","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.2609.29655","doi":"https://doi.org/10.48550/arxiv.2609.29655","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.29655","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:2609.29655","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29655","pdf_url":"https://arxiv.org/pdf/2609.29655","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":[{"id":"https://openalex.org/F4320325434","display_name":"Xiamen University","ror":"https://ror.org/00mcjh785"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214290495.pdf","grobid_xml":"https://content.openalex.org/works/W7214290495.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2021591064","https://openalex.org/W2952956902","https://openalex.org/W4255213956","https://openalex.org/W4404208797"],"related_works":["https://openalex.org/W7221796259","https://openalex.org/W7221754868","https://openalex.org/W1514105269","https://openalex.org/W7172526422","https://openalex.org/W7169808722","https://openalex.org/W2021340670","https://openalex.org/W7214368283","https://openalex.org/W4404208797","https://openalex.org/W7100777637","https://openalex.org/W1836917067"],"abstract_inverted_index":{"The":[0],"boundary":[1],"of":[2,18,120],"a":[3,15,45,109],"plane":[4],"amoeba":[5],"is":[6,14,66,89],"always":[7],"contained":[8],"in":[9],"its":[10],"contour,":[11],"and":[12,32,54,74,94,103,133],"equality":[13],"characteristic":[16],"feature":[17],"simple":[19,82],"Harnack":[20,83],"curves.":[21,84],"We":[22,69],"show":[23,104],"that":[24,77,105],"the":[25,64,106,114,118,126],"converse":[26],"fails,":[27],"even":[28],"under":[29],"strong":[30],"smoothness":[31],"nondegeneracy":[33],"assumptions.":[34],"For":[35],"every":[36],"two-dimensional":[37],"lattice":[38],"polygon,":[39],"except":[40],"unimodular":[41],"triangles,":[42],"we":[43],"construct":[44],"smooth":[46,50,55],"Newton-nondegenerate":[47],"curve":[48,65],"with":[49],"logarithmic":[51],"critical":[52,122],"locus":[53],"embedded":[56],"contour":[57,107],"satisfying":[58],"$\\mathcal":[59],"C(\\mathscr":[60],"A_f)=\\partial\\mathscr":[61],"A_f$,":[62],"although":[63],"not":[67,79,112],"Harnack.":[68],"also":[70],"provide":[71],"explicit":[72],"primitive":[73,87],"nonprimitive":[75],"families":[76],"are":[78],"torus-equivalent":[80],"to":[81],"A":[85],"concrete":[86],"example":[88],"certified":[90],"by":[91,130],"exact":[92],"elimination":[93],"Sturm":[95],"root":[96],"counting.":[97],"These":[98],"results":[99],"disprove":[100],"contour--boundary":[101],"rigidity":[102],"as":[108],"set":[110],"does":[111],"detect":[113],"real":[115],"structure":[116],"or":[117],"multiplicity":[119],"coincident":[121],"sheets,":[123],"thereby":[124],"refining":[125],"compensation":[127],"problem":[128],"proposed":[129],"Lang,":[131],"Shapiro,":[132],"Shustin.":[134]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-26T00:00:00"}