{"id":"https://openalex.org/W1971871539","doi":"https://doi.org/10.1090/s0025-5718-05-01729-1","title":"Solving quadratic equations using reduced unimodular quadratic forms","display_name":"Solving quadratic equations using reduced unimodular quadratic forms","publication_year":2005,"publication_date":"2005-01-27","ids":{"openalex":"https://openalex.org/W1971871539","doi":"https://doi.org/10.1090/s0025-5718-05-01729-1","mag":"1971871539"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-05-01729-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01729-1","pdf_url":"https://www.ams.org/mcom/2005-74-251/S0025-5718-05-01729-1/S0025-5718-05-01729-1.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/2005-74-251/S0025-5718-05-01729-1/S0025-5718-05-01729-1.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Denis Simon","orcid":null},"institutions":[{"id":"https://openalex.org/I98702875","display_name":"Universit\u00e9 de Caen Normandie","ror":"https://ror.org/051kpcy16","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210105918","https://openalex.org/I98702875"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Denis Simon","raw_affiliation_strings":["LMNO\u00e2\u0080\u0093UMR 6139, Universit\u00c3\u00a9 de Caen\u00e2\u0080\u0093Campus II, Bd du Mar\u00c3\u00a9chal Juin, BP 5186\u00e2\u0080\u009314032 Caen Cedex, France","LMNO\u2013UMR 6139, Universit\u00e9 de Caen\u2013Campus II, Bd du Mar\u00e9chal Juin, BP 5186\u201314032 Caen Cedex, France"],"affiliations":[{"raw_affiliation_string":"LMNO\u00e2\u0080\u0093UMR 6139, Universit\u00c3\u00a9 de Caen\u00e2\u0080\u0093Campus II, Bd du Mar\u00c3\u00a9chal Juin, BP 5186\u00e2\u0080\u009314032 Caen Cedex, France","institution_ids":[]},{"raw_affiliation_string":"LMNO\u2013UMR 6139, Universit\u00e9 de Caen\u2013Campus II, Bd du Mar\u00e9chal Juin, BP 5186\u201314032 Caen Cedex, France","institution_ids":["https://openalex.org/I98702875"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I98702875"],"apc_list":null,"apc_paid":null,"fwci":5.0513,"has_fulltext":true,"cited_by_count":49,"citation_normalized_percentile":{"value":0.95713943,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"74","issue":"251","first_page":"1531","last_page":"1543"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9997000098228455,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9990000128746033,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9962000250816345,"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/mathematics","display_name":"Mathematics","score":0.8436070680618286},{"id":"https://openalex.org/keywords/unimodular-matrix","display_name":"Unimodular matrix","score":0.828233003616333},{"id":"https://openalex.org/keywords/binary-quadratic-form","display_name":"Binary quadratic form","score":0.6949983239173889},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.6933360695838928},{"id":"https://openalex.org/keywords/isotropic-quadratic-form","display_name":"Isotropic quadratic form","score":0.681603193283081},{"id":"https://openalex.org/keywords/quartic-function","display_name":"Quartic function","score":0.6250790357589722},{"id":"https://openalex.org/keywords/ternary-operation","display_name":"Ternary operation","score":0.5471989512443542},{"id":"https://openalex.org/keywords/quadratic-function","display_name":"Quadratic function","score":0.5170524716377258},{"id":"https://openalex.org/keywords/quadratic-form","display_name":"Quadratic form (statistics)","score":0.49653321504592896},{"id":"https://openalex.org/keywords/discriminant","display_name":"Discriminant","score":0.48938173055648804},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4888383746147156},{"id":"https://openalex.org/keywords/definite-quadratic-form","display_name":"Definite quadratic form","score":0.47711923718452454},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.4755037724971771},{"id":"https://openalex.org/keywords/quadratic-programming","display_name":"Quadratic programming","score":0.47020038962364197},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4602104723453522},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.44097700715065},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.41787219047546387},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2920348048210144},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.274419367313385},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.23885953426361084},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.12726649641990662},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1187346875667572},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.06936594843864441}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8436070680618286},{"id":"https://openalex.org/C201958917","wikidata":"https://www.wikidata.org/wiki/Q1367055","display_name":"Unimodular matrix","level":2,"score":0.828233003616333},{"id":"https://openalex.org/C80695182","wikidata":"https://www.wikidata.org/wiki/Q864134","display_name":"Binary quadratic form","level":4,"score":0.6949983239173889},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.6933360695838928},{"id":"https://openalex.org/C21440807","wikidata":"https://www.wikidata.org/wiki/Q4968811","display_name":"Isotropic quadratic form","level":5,"score":0.681603193283081},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.6250790357589722},{"id":"https://openalex.org/C64452783","wikidata":"https://www.wikidata.org/wiki/Q1524945","display_name":"Ternary operation","level":2,"score":0.5471989512443542},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.5170524716377258},{"id":"https://openalex.org/C152022596","wikidata":"https://www.wikidata.org/wiki/Q7268361","display_name":"Quadratic form (statistics)","level":2,"score":0.49653321504592896},{"id":"https://openalex.org/C78397625","wikidata":"https://www.wikidata.org/wiki/Q192487","display_name":"Discriminant","level":2,"score":0.48938173055648804},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4888383746147156},{"id":"https://openalex.org/C104762803","wikidata":"https://www.wikidata.org/wiki/Q1182685","display_name":"Definite quadratic form","level":5,"score":0.47711923718452454},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.4755037724971771},{"id":"https://openalex.org/C81845259","wikidata":"https://www.wikidata.org/wiki/Q290117","display_name":"Quadratic programming","level":2,"score":0.47020038962364197},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4602104723453522},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.44097700715065},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.41787219047546387},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2920348048210144},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.274419367313385},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.23885953426361084},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.12726649641990662},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1187346875667572},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.06936594843864441},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1090/s0025-5718-05-01729-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01729-1","pdf_url":"https://www.ams.org/mcom/2005-74-251/S0025-5718-05-01729-1/S0025-5718-05-01729-1.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:10.1.1.98.6571","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.98.6571","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/2005-74-251/S0025-5718-05-01729-1/S0025-5718-05-01729-1.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-05-01729-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01729-1","pdf_url":"https://www.ams.org/mcom/2005-74-251/S0025-5718-05-01729-1/S0025-5718-05-01729-1.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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1971871539.pdf","grobid_xml":"https://content.openalex.org/works/W1971871539.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W1044193616","https://openalex.org/W1496801598","https://openalex.org/W1577641406","https://openalex.org/W1587415125","https://openalex.org/W1596713208","https://openalex.org/W1861663015","https://openalex.org/W1975287686","https://openalex.org/W1982264769","https://openalex.org/W1986401942","https://openalex.org/W2040541934","https://openalex.org/W4246973793"],"related_works":["https://openalex.org/W1559857115","https://openalex.org/W3006239198","https://openalex.org/W4387093665","https://openalex.org/W1990134970","https://openalex.org/W2963822715","https://openalex.org/W4206757637","https://openalex.org/W313220971","https://openalex.org/W2018752322","https://openalex.org/W223092903","https://openalex.org/W2352090799"],"abstract_inverted_index":{"Let":[0],"<inline-formula":[1,16,44,127,164,195,231,267],"content-type=\"math/mathml\">":[2,17,45,128,165,196,232,268],"<mml:math":[3,18,46,129,166,197,233,269],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[4,19,47,130,167,198,234,270],"alttext=\"upper":[5,271],"Q\">":[6,170,237,272],"<mml:semantics>":[7,23,53,141,171,201,238,273],"<mml:mi>Q</mml:mi>":[8,58,274],"<mml:annotation":[9,31,64,156,177,208,244,275],"encoding=\"application/x-tex\">Q</mml:annotation>":[10,276],"</mml:semantics>":[11,34,69,158,180,211,247,277],"</mml:math>":[12,35,70,159,181,212,248,278],"</inline-formula>":[13,36,71,160,182,213,249,279],"be":[14,187],"an":[15,113],"alttext=\"n":[20],"times":[21],"n\">":[22],"<mml:mrow>":[24,54,142,202],"<mml:mi>n</mml:mi>":[25,29],"<mml:mo>":[26,59,203],"\u00d7":[27],"</mml:mo>":[28,61,205],"</mml:mrow>":[30,63,155,176,207,243],"encoding=\"application/x-tex\">n\\times":[32],"n</mml:annotation>":[33],"symmetric":[37],"matrix":[38],"with":[39,43,98],"integral":[40],"entries":[41],"and":[42,251],"alttext=\"det":[48],"upper":[49,169,236],"Q":[50,66],"not-equals":[51],"0\">":[52,140],"<mml:mo":[55,144,151],"movablelimits=\"true\"":[56],"form=\"prefix\">det</mml:mo>":[57],"\u2260":[60],"<mml:mn>0</mml:mn>":[62,154],"encoding=\"application/x-tex\">\\det":[65],"\\neq":[67],"0</mml:annotation>":[68],",":[72,183,250],"but":[73],"not":[74],"necesarily":[75],"positive":[76],"definite.":[77],"We":[78,110],"describe":[79,112],"a":[80,119,124,184,254],"generalized":[81],"LLL":[82],"algorithm":[83,90,114,257],"to":[84,105,222],"reduce":[85],"this":[86,252],"quadratic":[87,94,121,125,191,228],"form.":[88],"This":[89],"either":[91],"reduces":[92],"the":[93,116,261,264],"form":[95],"or":[96],"stops":[97],"some":[99],"isotropic":[100],"vector.":[101],"It":[102],"is":[103,161,280],"proved":[104],"run":[106],"in":[107],"polynomial":[108,255],"time.":[109],"also":[111],"for":[115],"minimization":[117],"of":[118,193,217,263,266],"ternary":[120,227],"form:":[122],"when":[123],"equation":[126,192,229],"alttext=\"q":[131],"left-parenthesis":[132],"x":[133],"comma":[134,136],"y":[135],"z":[137],"right-parenthesis":[138],"equals":[139],"<mml:mi>q</mml:mi>":[143],"stretchy=\"false\">(</mml:mo>":[145],"<mml:mi>x</mml:mi>":[146],"<mml:mo>,</mml:mo>":[147,149],"<mml:mi>y</mml:mi>":[148],"<mml:mi>z</mml:mi>":[150],"stretchy=\"false\">)</mml:mo>":[152],"<mml:mo>=</mml:mo>":[153],"encoding=\"application/x-tex\">q(x,y,z)=0</mml:annotation>":[157],"solvable":[162],"over":[163,230],"alttext=\"double-struck":[168,235],"<mml:mrow":[172,239],"class=\"MJX-TeXAtom-ORD\">":[173,240],"<mml:mi":[174,241],"mathvariant=\"double-struck\">Q</mml:mi>":[175,242],"encoding=\"application/x-tex\">\\mathbb":[178,245],"{Q}</mml:annotation>":[179,246],"solution":[185],"can":[186],"deduced":[188],"from":[189],"another":[190],"determinant":[194,265],"alttext=\"plus-or-minus":[199],"1\">":[200],"\u00b1":[204],"<mml:mn>1</mml:mn>":[206],"encoding=\"application/x-tex\">\\pm":[209],"1</mml:annotation>":[210],".":[214],"The":[215],"combination":[216],"these":[218],"algorithms":[219],"allows":[220],"us":[221],"solve":[223],"efficiently":[224],"any":[225],"general":[226],"gives":[253],"time":[256],"(as":[258],"soon":[259],"as":[260],"factorization":[262],"known).":[281]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-22T06:01:30.510260","created_date":"2025-10-10T00:00:00"}
