{"id":"https://openalex.org/W7214186408","doi":"https://doi.org/10.48550/arxiv.2609.27049","title":"Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+","display_name":"Linear Algebra and Algebraic Geometry: A Matrix Construction for Classifying Families of Algebraic Curves with No Solutions in Z^+","publication_year":2026,"publication_date":"2026-09-22","ids":{"openalex":"https://openalex.org/W7214186408","doi":"https://doi.org/10.48550/arxiv.2609.27049"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.27049","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27049","pdf_url":"https://arxiv.org/pdf/2609.27049","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/2609.27049","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5092668818","display_name":"Shazali Abdalla Fadul","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I224517680","display_name":"Al-Neelain University","ror":"https://ror.org/05dvsnx49","country_code":"SD","type":"education","lineage":["https://openalex.org/I224517680"]}],"countries":["SD"],"is_corresponding":true,"raw_author_name":"Fadul, Shazali Abdalla","raw_affiliation_strings":["Faculty Of Mathematics Sciences & Statistics , AL-neelain University , Khartoum , Sudan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty Of Mathematics Sciences & Statistics , AL-neelain University , Khartoum , Sudan","institution_ids":["https://openalex.org/I224517680"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5092668818"],"corresponding_institution_ids":["https://openalex.org/I224517680"],"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9994999766349792,"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":0.9994999766349792,"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/T11166","display_name":"Analytic Number Theory Research","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/algebraic-curves","display_name":"algebraic curves","score":0.8050000071525574},{"id":"https://openalex.org/keywords/lattice-points","display_name":"lattice points","score":0.44999998807907104},{"id":"https://openalex.org/keywords/curve-classification","display_name":"classification of curves","score":0.6320000290870667},{"id":"https://openalex.org/keywords/nonsingular-curves","display_name":"nonsingular curves","score":0.7300000190734863},{"id":"https://openalex.org/keywords/elliptic-curves","display_name":"elliptic curves","score":0.7480000257492065},{"id":"https://openalex.org/keywords/mordell-weil-theorem","display_name":"Mordell-Weil theorem","score":0.6240000128746033},{"id":"https://openalex.org/keywords/faltings-theorem","display_name":"Faltings' theorem","score":0.656000018119812},{"id":"https://openalex.org/keywords/matrix-construction","display_name":"matrix construction","score":0.4000000059604645}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8357999920845032},{"id":"https://openalex.org/C207043602","wikidata":"https://www.wikidata.org/wiki/Q266237","display_name":"Algebraic curve","level":2,"score":0.6640999913215637},{"id":"https://openalex.org/C2731732","wikidata":"https://www.wikidata.org/wiki/Q7301119","display_name":"Real algebraic geometry","level":3,"score":0.6531999707221985},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.6175000071525574},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5734000205993652},{"id":"https://openalex.org/C69653121","wikidata":"https://www.wikidata.org/wiki/Q1434805","display_name":"Algebraic surface","level":3,"score":0.5525000095367432},{"id":"https://openalex.org/C25971838","wikidata":"https://www.wikidata.org/wiki/Q5277261","display_name":"Dimension of an algebraic variety","level":3,"score":0.5205000042915344},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.5138000249862671},{"id":"https://openalex.org/C68363185","wikidata":"https://www.wikidata.org/wiki/Q180969","display_name":"Algebraic geometry","level":2,"score":0.48730000853538513},{"id":"https://openalex.org/C53622353","wikidata":"https://www.wikidata.org/wiki/Q1518913","display_name":"Family of curves","level":2,"score":0.48510000109672546},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.48429998755455017},{"id":"https://openalex.org/C10996884","wikidata":"https://www.wikidata.org/wiki/Q5508773","display_name":"Function field of an algebraic variety","level":3,"score":0.4731000065803528},{"id":"https://openalex.org/C201482947","wikidata":"https://www.wikidata.org/wiki/Q7524253","display_name":"Singular point of an algebraic variety","level":5,"score":0.4618000090122223},{"id":"https://openalex.org/C199793520","wikidata":"https://www.wikidata.org/wiki/Q4723996","display_name":"Algebraic cycle","level":3,"score":0.460099995136261},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.45320001244544983},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.42739999294281006},{"id":"https://openalex.org/C139352143","wikidata":"https://www.wikidata.org/wiki/Q82571","display_name":"Linear algebra","level":2,"score":0.4271000027656555},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.41909998655319214},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.4043000042438507},{"id":"https://openalex.org/C26959085","wikidata":"https://www.wikidata.org/wiki/Q5275330","display_name":"Differential algebraic geometry","level":5,"score":0.37709999084472656},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35580000281333923},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.3458999991416931},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.3361999988555908},{"id":"https://openalex.org/C554573382","wikidata":"https://www.wikidata.org/wiki/Q4723988","display_name":"Algebraic analysis","level":5,"score":0.328000009059906},{"id":"https://openalex.org/C184363387","wikidata":"https://www.wikidata.org/wiki/Q3001219","display_name":"Stable curve","level":4,"score":0.31360000371932983},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.30320000648498535},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.30070000886917114},{"id":"https://openalex.org/C148607811","wikidata":"https://www.wikidata.org/wiki/Q746863","display_name":"Algebraic function","level":3,"score":0.2897000014781952},{"id":"https://openalex.org/C157369684","wikidata":"https://www.wikidata.org/wiki/Q34740","display_name":"Genus","level":2,"score":0.2896000146865845},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.2872999906539917},{"id":"https://openalex.org/C173832452","wikidata":"https://www.wikidata.org/wiki/Q162608","display_name":"Algebraic element","level":5,"score":0.27219998836517334},{"id":"https://openalex.org/C165761256","wikidata":"https://www.wikidata.org/wiki/Q648995","display_name":"Algebraic variety","level":3,"score":0.2639000117778778},{"id":"https://openalex.org/C144559511","wikidata":"https://www.wikidata.org/wiki/Q2986279","display_name":"Principal (computer security)","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.27049","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27049","pdf_url":"https://arxiv.org/pdf/2609.27049","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.2609.27049","doi":"https://doi.org/10.48550/arxiv.2609.27049","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.27049","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:2609.27049","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27049","pdf_url":"https://arxiv.org/pdf/2609.27049","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/W7214186408.pdf","grobid_xml":"https://content.openalex.org/works/W7214186408.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W1604636366","https://openalex.org/W1970080688","https://openalex.org/W2011844852","https://openalex.org/W2038383731","https://openalex.org/W2073082930","https://openalex.org/W7214021526"],"related_works":["https://openalex.org/W7206154475","https://openalex.org/W4244121893","https://openalex.org/W4205642002","https://openalex.org/W2014057071","https://openalex.org/W2288733545","https://openalex.org/W2236129831","https://openalex.org/W2912712966","https://openalex.org/W2020258451","https://openalex.org/W7099182947","https://openalex.org/W4255601932"],"abstract_inverted_index":{"A":[0],"substantial":[1],"body":[2],"of":[3,9,18,35,72,85,146,161,169,174],"results":[4,30,42],"is":[5,76],"known":[6],"for":[7,31,53,82],"curves":[8,34,56,87,160,167],"degree":[10,36,162],"$d=3$,":[11],"namely":[12],"elliptic":[13],"curves,":[14],"including":[15],"the":[16,73,143,147,175,198],"theorems":[17],"Siegel,":[19],"Mazur,":[20],"and":[21,102,116,137,151,185,196],"Mordell,":[22],"among":[23],"others.":[24],"More":[25],"generally,":[26],"Faltings":[27],"established":[28],"comprehensive":[29],"all":[32],"algebraic":[33,55,86,100,159],"$d":[37],"\\ge":[38,172,180],"2$.":[39],"However,":[40],"these":[41],"do":[43],"not,":[44],"in":[45,91,113],"general,":[46],"furnish":[47],"an":[48],"explicit":[49],"or":[50],"systematic":[51],"procedure":[52,111],"classifying":[54,83],"that":[57],"fail":[58],"to":[59,78],"admit":[60],"solutions":[61,90],"over":[62],"a":[63,80,97,109,122,155],"prescribed":[64],"set":[65,145],"such":[66,154],"as":[67,211],"$\\mathbb{Z}^{+}$.":[68,92],"The":[69],"principal":[70],"aim":[71],"present":[74],"work":[75],"accordingly":[77],"develop":[79,108],"method":[81],"families":[84],"having":[88],"no":[89],"We":[93],"begin":[94],"by":[95],"establishing":[96],"correspondence":[98],"between":[99],"geometry":[101],"linear":[103,123],"algebra,":[104],"from":[105],"which":[106],"we":[107,157],"classification":[110],"grounded":[112],"linear-algebraic":[114],"constructions":[115],"tools.":[117],"Specifically,":[118],"let":[119,138],"$S:Ax=b$":[120],"be":[121],"system,":[124,148],"where":[125],"$A":[126],"\\in":[127,187],"M_{3k":[128],"\\times":[129],"m}(\\mathbb{Z})$":[130],"satisfies":[131],"suitable":[132],"conditions":[133],"on":[134],"its":[135],"entries,":[136],"$S=\\{s_{1},s_{2},\\dots,s_{N}\\}":[139],"\\subset":[140],"\\mathbb{Z}^{m}$":[141],"denote":[142],"solution":[144,156],"with":[149],"$A=(a_{ij})$,":[150],"$s_{i}=(c_{1},c_{2},\\dots,c_{n+1})$.":[152],"From":[153],"construct":[158],"$n":[163],"\\le":[164,192,194,207,209],"m$,":[165],"nonsingular":[166],"$C_{js_i}$":[168],"genus":[170],"$g":[171],"0$,":[173],"form":[176],"$C_{js_{i}}:Y^{2}=a_{3j1}c_{1}X^{n}+a_{3j2}c_{2}X^{n-1}+\\dots+a_{3jn}c_{n}$.":[177],"If":[178],"$Y":[179],"1,":[181],"X":[182],">":[183],"1$":[184],"$\\forall(X,Y)":[186],"C_{js_{i}}(\\mathbb{Z}^{+})$":[188],"holds":[189],"then":[190,205],"$[C_{js_{i}}(\\mathbb{Z}^{+})]_{1":[191,206],"j":[193,208],"k}^{N}=\\emptyset$":[195],"if":[197],"system":[199],"$S$":[200],"admits":[201],"infinitely":[202],"many":[203],"solutions,":[204],"k}^{\\infty}=\\emptyset$":[210],"$N":[212],"\\longrightarrow":[213],"\\infty$.":[214]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-25T00:00:00"}