{"id":"https://openalex.org/W7203886500","doi":"https://doi.org/10.48550/arxiv.2608.19112","title":"On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems","display_name":"On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems","publication_year":2026,"publication_date":"2026-08-19","ids":{"openalex":"https://openalex.org/W7203886500","doi":"https://doi.org/10.48550/arxiv.2608.19112"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.19112","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.19112","pdf_url":"https://arxiv.org/pdf/2608.19112","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/2608.19112","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5147685720","display_name":"Hanwen Liu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I39555362","display_name":"University of Warwick","ror":"https://ror.org/01a77tt86","country_code":"GB","type":"education","lineage":["https://openalex.org/I39555362"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Liu, Hanwen","raw_affiliation_strings":["Mathematics Institute , University of Warwick"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics Institute , University of Warwick","institution_ids":["https://openalex.org/I39555362"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5147685720"],"corresponding_institution_ids":["https://openalex.org/I39555362"],"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":0.8647000193595886,"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":0.8647000193595886,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.05999999865889549,"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":0.010700000450015068,"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/lorentzian-signature","display_name":"Lorentzian signature","score":0.890999972820282},{"id":"https://openalex.org/keywords/monge-ampere-equation","display_name":"Monge-Amp\u00e8re equation","score":0.8809999823570251},{"id":"https://openalex.org/keywords/polynomial-automorphisms","display_name":"polynomial automorphisms","score":0.8790000081062317},{"id":"https://openalex.org/keywords/jacobian-conjecture","display_name":"Jacobian conjecture","score":0.8309999704360962},{"id":"https://openalex.org/keywords/legendre-transform","display_name":"Legendre transform","score":0.8629999756813049}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8582000136375427},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.6571999788284302},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.6025999784469604},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.597599983215332},{"id":"https://openalex.org/C118712358","wikidata":"https://www.wikidata.org/wiki/Q782566","display_name":"Automorphism","level":2,"score":0.5965999960899353},{"id":"https://openalex.org/C17103678","wikidata":"https://www.wikidata.org/wiki/Q1474074","display_name":"Homogeneous polynomial","level":4,"score":0.5889000296592712},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5410000085830688},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5145000219345093},{"id":"https://openalex.org/C4042151","wikidata":"https://www.wikidata.org/wiki/Q215111","display_name":"Homomorphism","level":2,"score":0.5077000260353088},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.454800009727478},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.42809998989105225},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.3898000121116638},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.36910000443458557},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3634999990463257},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3334999978542328},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.33329999446868896},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.325300008058548},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3246000111103058},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.3231000006198883},{"id":"https://openalex.org/C42321380","wikidata":"https://www.wikidata.org/wiki/Q849705","display_name":"Characteristic polynomial","level":3,"score":0.29809999465942383},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.28780001401901245},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.27730000019073486},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.2678999900817871},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.2615000009536743},{"id":"https://openalex.org/C135981907","wikidata":"https://www.wikidata.org/wiki/Q188056","display_name":"Triangulation","level":2,"score":0.25929999351501465},{"id":"https://openalex.org/C171036898","wikidata":"https://www.wikidata.org/wiki/Q256355","display_name":"Equivariant map","level":2,"score":0.25609999895095825},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.2513999938964844},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.19112","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.19112","pdf_url":"https://arxiv.org/pdf/2608.19112","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.2608.19112","doi":"https://doi.org/10.48550/arxiv.2608.19112","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.19112","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:2608.19112","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.19112","pdf_url":"https://arxiv.org/pdf/2608.19112","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/F4320320279","display_name":"University of Warwick","ror":"https://ror.org/01a77tt86"},{"id":"https://openalex.org/F4320320915","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052"},{"id":"https://openalex.org/F4320322605","display_name":"Ningbo University","ror":"https://ror.org/03et85d35"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7203886500.pdf","grobid_xml":"https://content.openalex.org/works/W7203886500.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W1583189483","https://openalex.org/W2015501897","https://openalex.org/W2076634702","https://openalex.org/W2081461845","https://openalex.org/W2103884324","https://openalex.org/W2113547185","https://openalex.org/W2319067800","https://openalex.org/W2520639029","https://openalex.org/W4210565893","https://openalex.org/W4387448604","https://openalex.org/W7171441141"],"related_works":["https://openalex.org/W7203618291","https://openalex.org/W7171441141","https://openalex.org/W2113547185","https://openalex.org/W2076634702","https://openalex.org/W7171740776","https://openalex.org/W2103884324","https://openalex.org/W2967472918","https://openalex.org/W2520639029","https://openalex.org/W2953017695","https://openalex.org/W2413884641"],"abstract_inverted_index":{"As":[0,180],"a":[1,21,31,62,81,88,113,116,121,154,164,174,190,214],"close":[2],"relative":[3],"of":[4,20,109,157,173,240],"the":[5,8,16,25,49,54,95,106,139,160,171,186,194,210,230],"Jacobian":[6],"conjecture,":[7],"Hessian":[9,36,56,71,231],"conjecture":[10,37,57,232],"in":[11,43,58,66,233],"dimension":[12,199],"$n$":[13],"states":[14],"that":[15,94,105,120,170,185,206],"local":[17],"Legendre":[18],"transform":[19],"polynomial":[22,32,63,117,191,238],"solution":[23],"to":[24,86,178],"Monge--Amp\u00e8re":[26],"equation":[27],"$\\det(\\operatorname{Hess}(\u03d5))=\\pm1$":[28],"is":[29,38,100,115,176,189],"also":[30,204],"solution.":[33],"The":[34],"general":[35],"false":[39],"for":[40,84,225,236],"$n\\geq5$,":[41],"while":[42],"Riemannian":[44],"signature":[45],"it":[46],"follows":[47],"from":[48],"J\u00f6rgens--Calabi--Pogorelov":[50],"theorem.":[51],"We":[52,102,168,203],"study":[53],"four-dimensional":[55],"Lorentzian":[59],"signature.":[60],"For":[61],"potential":[64,111,153,211,239],"$\u03d5$":[65,85,126,212],"four":[67],"real":[68],"variables":[69],"whose":[70],"matrix":[72],"has":[73,197],"index":[74],"$1$":[75],"and":[76,119,146,163],"determinant":[77],"$-1$,":[78],"we":[79,137,149,183,218,228],"define":[80],"constant":[82,90],"pivot":[83,114,122,175],"be":[87],"nonzero":[89],"vector":[91],"$\u03be$":[92],"such":[93],"second":[96],"directional":[97],"derivative":[98],"$D_\u03be^2\u03d5$":[99],"constant.":[101],"then":[103,209,219],"prove":[104,138,169,184,229],"gradient":[107,187],"mapping":[108,188],"every":[110,237],"admitting":[112],"automorphism,":[118],"always":[123],"exists":[124],"when":[125,142],"decomposes":[127],"into":[128],"homogeneous":[129],"pieces":[130],"as":[131],"$\u03d5=\u03d5_d+\u03d5_{d-1}+\u03d5_2+\u03d5_1+\u03d5_0$":[132],"with":[133,151],"$d\\geq4$.":[134],"More":[135],"generally,":[136],"same":[140],"conclusion":[141],"$$\u03d5=\u03d5_d+\\cdots+\u03d5_{d-k}+\u03d5_2+\u03d5_1+\u03d5_0,$$":[143],"where":[144],"$k\\geq0$":[145],"$d\\geq4k+3$.":[147],"Then,":[148],"associate":[150],"each":[152],"linear":[155],"system":[156,196],"quadrics,":[158],"called":[159],"Hesse":[161,195],"system,":[162],"canonical":[165],"homomorphism":[166],"$\u03bc_\u03d5$.":[167],"existence":[172],"equivalent":[177],"$\\operatorname{rank}(\u03bc_\u03d5)\\leq55$.":[179],"an":[181,221],"application,":[182],"automorphism":[192],"whenever":[193],"complex":[198],"at":[200,242],"most":[201,243],"4.":[202],"show":[205],"if":[207],"${\\det(\\operatorname{Hess}(\u03d5-\u03d5_2))\\equiv0}$,":[208],"admits":[213],"pivot.":[215],"After":[216],"that,":[217],"give":[220],"analytic":[222],"degeneracy":[223],"criterion":[224],"$\\operatorname{Hess}(\u03d5-\u03d5_2)$.":[226],"Finally,":[227],"this":[234],"setting":[235],"degree":[241],"five.":[244]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-22T00:00:00"}