{"id":"https://openalex.org/W7204163554","doi":"https://doi.org/10.48550/arxiv.2608.22049","title":"How many points does an affine algebraic set have in residue classes modulo n ?","display_name":"How many points does an affine algebraic set have in residue classes modulo n ?","publication_year":2026,"publication_date":"2026-08-22","ids":{"openalex":"https://openalex.org/W7204163554","doi":"https://doi.org/10.48550/arxiv.2608.22049"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.22049","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22049","pdf_url":"https://arxiv.org/pdf/2608.22049","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.22049","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044551951","display_name":"Mihai Prunescu","orcid":"https://orcid.org/0000-0003-1627-050X","observed_orcids":["https://orcid.org/0000-0003-1627-050X"]},"institutions":[{"id":"https://openalex.org/I141595442","display_name":"University of Bucharest","ror":"https://ror.org/02x2v6p15","country_code":"RO","type":"education","lineage":["https://openalex.org/I141595442"]},{"id":"https://openalex.org/I4405262991","display_name":"IMAR - Institutul de Matematic\u0103 \u201eSimion Stoilow\u201d al Academiei Rom\u00e2ne","ror":"https://ror.org/0122mh736","country_code":"RO","type":"education","lineage":["https://openalex.org/I4405262991","https://openalex.org/I58077936"]}],"countries":["RO"],"is_corresponding":true,"raw_author_name":"Prunescu, Mihai","raw_affiliation_strings":["Research Center for Logic, Optimization and Security (LOS) , Faculty of Mathematics and Computer Science , University of Bucharest , Academiei 14 , Bucharest ( RO-010014) , Romania","Simion Stoilow Institute of Mathematics of the Romanian Academy , Research unit 5 , P. O. Box 1-764 , Bucharest ( RO-014700) , Romania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Research Center for Logic, Optimization and Security (LOS) , Faculty of Mathematics and Computer Science , University of Bucharest , Academiei 14 , Bucharest ( RO-010014) , Romania","institution_ids":["https://openalex.org/I141595442"]},{"raw_affiliation_string":"Simion Stoilow Institute of Mathematics of the Romanian Academy , Research unit 5 , P. O. Box 1-764 , Bucharest ( RO-014700) , Romania","institution_ids":["https://openalex.org/I4405262991"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5044551951"],"corresponding_institution_ids":["https://openalex.org/I141595442","https://openalex.org/I4405262991"],"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":0.9208999872207642,"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":0.9208999872207642,"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.03889999911189079,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"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/T11166","display_name":"Analytic Number Theory Research","score":0.028300000354647636,"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/elliptic-curves","display_name":"elliptic curves","score":0.8429999947547913},{"id":"https://openalex.org/keywords/division-with-remainder","display_name":"division with remainder","score":0.7570000290870667}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8565999865531921},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.551800012588501},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5307999849319458},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.46799999475479126},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.46230000257492065},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.435699999332428},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.4009000062942505},{"id":"https://openalex.org/C167315692","wikidata":"https://www.wikidata.org/wiki/Q3622729","display_name":"Affine arithmetic","level":3,"score":0.3935000002384186},{"id":"https://openalex.org/C39613435","wikidata":"https://www.wikidata.org/wiki/Q846677","display_name":"Remainder","level":2,"score":0.3693999946117401},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.358599990606308},{"id":"https://openalex.org/C164810661","wikidata":"https://www.wikidata.org/wiki/Q1358313","display_name":"Ring of integers","level":3,"score":0.35249999165534973},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.34150001406669617},{"id":"https://openalex.org/C23684344","wikidata":"https://www.wikidata.org/wiki/Q382520","display_name":"Affine geometry","level":4,"score":0.34040001034736633},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3395000100135803},{"id":"https://openalex.org/C112128483","wikidata":"https://www.wikidata.org/wiki/Q382504","display_name":"Affine hull","level":4,"score":0.33169999718666077},{"id":"https://openalex.org/C121444067","wikidata":"https://www.wikidata.org/wiki/Q2835817","display_name":"Schoof's algorithm","level":4,"score":0.3098999857902527},{"id":"https://openalex.org/C157567686","wikidata":"https://www.wikidata.org/wiki/Q7644139","display_name":"Supersingular elliptic curve","level":3,"score":0.3093000054359436},{"id":"https://openalex.org/C207221997","wikidata":"https://www.wikidata.org/wiki/Q938614","display_name":"Affine combination","level":3,"score":0.2969000041484833},{"id":"https://openalex.org/C151635116","wikidata":"https://www.wikidata.org/wiki/Q193878","display_name":"Chinese remainder theorem","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C87905105","wikidata":"https://www.wikidata.org/wiki/Q5284582","display_name":"Division polynomials","level":5,"score":0.2858999967575073},{"id":"https://openalex.org/C154610627","wikidata":"https://www.wikidata.org/wiki/Q382491","display_name":"Affine plane (incidence geometry)","level":3,"score":0.28540000319480896},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28279998898506165},{"id":"https://openalex.org/C2731732","wikidata":"https://www.wikidata.org/wiki/Q7301119","display_name":"Real algebraic geometry","level":3,"score":0.27320000529289246},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.26339998841285706},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.26249998807907104},{"id":"https://openalex.org/C25971838","wikidata":"https://www.wikidata.org/wiki/Q5277261","display_name":"Dimension of an algebraic variety","level":3,"score":0.25859999656677246},{"id":"https://openalex.org/C185579658","wikidata":"https://www.wikidata.org/wiki/Q3554813","display_name":"Affine variety","level":3,"score":0.2565999925136566},{"id":"https://openalex.org/C199793520","wikidata":"https://www.wikidata.org/wiki/Q4723996","display_name":"Algebraic cycle","level":3,"score":0.25040000677108765}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.22049","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22049","pdf_url":"https://arxiv.org/pdf/2608.22049","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.2608.22049","doi":"https://doi.org/10.48550/arxiv.2608.22049","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.22049","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:2608.22049","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.22049","pdf_url":"https://arxiv.org/pdf/2608.22049","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":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204163554.pdf","grobid_xml":"https://content.openalex.org/works/W7204163554.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7211905508","https://openalex.org/W7023659020","https://openalex.org/W4221113096","https://openalex.org/W7049276148","https://openalex.org/W2795442031","https://openalex.org/W2296401598","https://openalex.org/W7096889481","https://openalex.org/W2086441423","https://openalex.org/W2078436079","https://openalex.org/W2317188011"],"abstract_inverted_index":{"We":[0],"show":[1],"that":[2],"for":[3,44],"every":[4],"uniform":[5],"family":[6,20],"of":[7,18,27,60],"affine":[8],"algebraic":[9],"sets,":[10],"there":[11],"is":[12],"a":[13,42],"formula":[14,43],"in":[15,22,30,48],"the":[16,19,25,28,31,45],"parameters":[17],"and":[21,67],"$n$,":[23],"expressing":[24],"cardinality":[26],"set":[29],"ring":[32],"$\\mathbb":[33],"Z/n":[34],"\\mathbb":[35],"Z$.":[36],"In":[37],"particular,":[38],"we":[39],"construct":[40],"such":[41],"elliptic":[46],"curves":[47],"Weierstrass":[49],"normal":[50],"form.":[51],"These":[52],"formulas":[53],"are":[54],"arithmetic":[55],"terms:":[56],"fixed":[57],"finite":[58],"compositions":[59],"additions,":[61],"multiplications,":[62],"subtractions,":[63],"divisions":[64],"with":[65],"remainder":[66],"integer":[68],"exponentiations.":[69]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-08-26T00:00:00"}