{"id":"https://openalex.org/W2054332045","doi":"https://doi.org/10.1090/s0025-5718-02-01455-2","title":"A mass formula for unimodular lattices with no roots","display_name":"A mass formula for unimodular lattices with no roots","publication_year":2002,"publication_date":"2002-06-25","ids":{"openalex":"https://openalex.org/W2054332045","doi":"https://doi.org/10.1090/s0025-5718-02-01455-2","mag":"2054332045"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-02-01455-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-02-01455-2","pdf_url":"https://www.ams.org/mcom/2003-72-242/S0025-5718-02-01455-2/S0025-5718-02-01455-2.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2003-72-242/S0025-5718-02-01455-2/S0025-5718-02-01455-2.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085529697","display_name":"Oliver D. King","orcid":"https://orcid.org/0000-0002-9460-4491"},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]},{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Oliver King","raw_affiliation_strings":["Department of Biological Chemistry and Molecular Pharmacology, Harvard Medical School, 250 Longwood Avenue, SGMB-322, Boston, Massachusetts 02115","Department of Mathematics, University of California, Berkeley, California 94720"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Biological Chemistry and Molecular Pharmacology, Harvard Medical School, 250 Longwood Avenue, SGMB-322, Boston, Massachusetts 02115","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Department of Mathematics, University of California, Berkeley, California 94720","institution_ids":["https://openalex.org/I95457486"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5085529697"],"corresponding_institution_ids":["https://openalex.org/I136199984","https://openalex.org/I95457486"],"apc_list":null,"apc_paid":null,"fwci":3.262,"has_fulltext":true,"cited_by_count":58,"citation_normalized_percentile":{"value":0.91690141,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"72","issue":"242","first_page":"839","last_page":"863"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9878000020980835,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9696000218391418,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/unimodular-matrix","display_name":"Unimodular matrix","score":0.7415102124214172},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5867753624916077},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5479649305343628},{"id":"https://openalex.org/keywords/annotation","display_name":"Annotation","score":0.4799671173095703},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4792923331260681},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.41093334555625916},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.36795711517333984},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3436684012413025},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.19189241528511047},{"id":"https://openalex.org/keywords/geology","display_name":"Geology","score":0.07633158564567566}],"concepts":[{"id":"https://openalex.org/C201958917","wikidata":"https://www.wikidata.org/wiki/Q1367055","display_name":"Unimodular matrix","level":2,"score":0.7415102124214172},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5867753624916077},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5479649305343628},{"id":"https://openalex.org/C2776321320","wikidata":"https://www.wikidata.org/wiki/Q857525","display_name":"Annotation","level":2,"score":0.4799671173095703},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4792923331260681},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.41093334555625916},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.36795711517333984},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3436684012413025},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.19189241528511047},{"id":"https://openalex.org/C127313418","wikidata":"https://www.wikidata.org/wiki/Q1069","display_name":"Geology","level":0,"score":0.07633158564567566},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1090/s0025-5718-02-01455-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-02-01455-2","pdf_url":"https://www.ams.org/mcom/2003-72-242/S0025-5718-02-01455-2/S0025-5718-02-01455-2.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.15.1951","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.1951","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://csua.berkeley.edu/~ok/mcom-final.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.153.7476","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.153.7476","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/mcom/2003-72-242/S0025-5718-02-01455-2/S0025-5718-02-01455-2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.23.8303","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.8303","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://csua.berkeley.edu/~ok/massform.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.233.8227","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.233.8227","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0012231v2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.234.2187","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.234.2187","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0012231v3.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.235.4047","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.235.4047","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0012231v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.26.9857","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.9857","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.berkeley.edu/~king/massform.ps","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-02-01455-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-02-01455-2","pdf_url":"https://www.ams.org/mcom/2003-72-242/S0025-5718-02-01455-2/S0025-5718-02-01455-2.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320006","display_name":"Royal Society","ror":"https://ror.org/03wnrjx87"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2054332045.pdf","grobid_xml":"https://content.openalex.org/works/W2054332045.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W37622074","https://openalex.org/W337358633","https://openalex.org/W976251535","https://openalex.org/W1484321265","https://openalex.org/W1503875804","https://openalex.org/W1544039992","https://openalex.org/W1585954501","https://openalex.org/W1591068227","https://openalex.org/W1645108173","https://openalex.org/W1831538018","https://openalex.org/W1983655075","https://openalex.org/W1985849796","https://openalex.org/W1988703714","https://openalex.org/W1995990042","https://openalex.org/W2000054582","https://openalex.org/W2011039300","https://openalex.org/W2028378109","https://openalex.org/W2051539504","https://openalex.org/W2065933899","https://openalex.org/W2070788983","https://openalex.org/W2078008217","https://openalex.org/W2085285043","https://openalex.org/W2104540562","https://openalex.org/W2152441705","https://openalex.org/W2159597320","https://openalex.org/W2172094104","https://openalex.org/W2317183192","https://openalex.org/W2318363093","https://openalex.org/W2328100090","https://openalex.org/W2752885492","https://openalex.org/W2964088494","https://openalex.org/W2995339545","https://openalex.org/W4239679971","https://openalex.org/W4246435416","https://openalex.org/W4292408570","https://openalex.org/W6625450870","https://openalex.org/W6647659499"],"related_works":["https://openalex.org/W2011745869","https://openalex.org/W2088028898","https://openalex.org/W2950905816","https://openalex.org/W211627054","https://openalex.org/W1608437562","https://openalex.org/W2083668208","https://openalex.org/W3141916192","https://openalex.org/W2171197936","https://openalex.org/W2094499544","https://openalex.org/W2062949095"],"abstract_inverted_index":{"We":[0,26,137],"derive":[1],"a":[2],"mass":[3,84,95,161],"formula":[4,29],"for":[5,30,42],"<inline-formula":[6,57,105],"content-type=\"math/mathml\">":[7,58,106],"<mml:math":[8,59,107],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[9,60,108],"alttext=\"n\">":[10],"<mml:semantics>":[11,64,112],"<mml:mi>n</mml:mi>":[12,66,114],"<mml:annotation":[13,72,120],"encoding=\"application/x-tex\">n</mml:annotation>":[14],"</mml:semantics>":[15,75,123],"</mml:math>":[16,76,124],"</inline-formula>":[17,77,125],"-dimensional":[18],"unimodular":[19,48,53,87,98,148],"lattices":[20,50,54,89,99,149],"having":[21],"any":[22],"prescribed":[23],"root":[24,44],"system.":[25],"use":[27],"Katsurada\u2019s":[28],"the":[31,83,94,144,159],"Fourier":[32],"coefficients":[33],"of":[34,46,55,85,96,146],"Siegel":[35],"Eisenstein":[36],"series":[37],"to":[38,153],"compute":[39,139],"these":[40],"masses":[41],"all":[43],"systems":[45],"even":[47,86],"32-dimensional":[49,88],"and":[51,93,129,135],"odd":[52,97],"dimension":[56,104],"alttext=\"n":[61,109],"less-than-or-equal-to":[62,110],"30\">":[63,111],"<mml:mrow>":[65,113],"<mml:mo>":[67,115],"\u2264":[68,116],"</mml:mo>":[69,117],"<mml:mn>30</mml:mn>":[70,118],"</mml:mrow>":[71,119],"encoding=\"application/x-tex\">n\\leq":[73,121],"30</mml:annotation>":[74,122],".":[78],"In":[79],"particular,":[80],"we":[81],"find":[82],"with":[90,100],"no":[91,101],"roots,":[92],"roots":[102],"in":[103,132,150],",":[126],"verifying":[127],"Bacher":[128],"Venkov\u2019s":[130],"enumerations":[131],"dimensions":[133,151],"27":[134],"28.":[136],"also":[138],"better":[140],"lower":[141],"bounds":[142],"on":[143],"number":[145],"inequivalent":[147],"26":[152],"30":[154],"than":[155],"those":[156],"afforded":[157],"by":[158],"Minkowski-Siegel":[160],"constants.":[162]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":5},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":9},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-12T08:23:45.883708","created_date":"2025-10-10T00:00:00"}
