{"id":"https://openalex.org/W2161310718","doi":"https://doi.org/10.1145/1823931.1823934","title":"Unit normalization of multinomials over Gaussian integers","display_name":"Unit normalization of multinomials over Gaussian integers","publication_year":2010,"publication_date":"2010-06-24","ids":{"openalex":"https://openalex.org/W2161310718","doi":"https://doi.org/10.1145/1823931.1823934","mag":"2161310718"},"language":"en","primary_location":{"id":"doi:10.1145/1823931.1823934","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1823931.1823934","pdf_url":null,"source":{"id":"https://openalex.org/S176185422","display_name":"ACM communications in computer algebra","issn_l":"1932-2232","issn":["1932-2232","1932-2240"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320725","host_organization_name":"SIGSAM","host_organization_lineage":["https://openalex.org/P4310320725"],"host_organization_lineage_names":["SIGSAM"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Communications in Computer Algebra","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038500480","display_name":"David R. Stoutemyer","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"David R. Stoutemyer","raw_affiliation_strings":[""],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5038500480"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3432,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.6534789,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"43","issue":"3/4","first_page":"73","last_page":"76"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9983000159263611,"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":0.9983000159263611,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9975000023841858,"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/T11034","display_name":"Digital Filter Design and Implementation","score":0.977400004863739,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/normalization","display_name":"Normalization (sociology)","score":0.8181967735290527},{"id":"https://openalex.org/keywords/unique-factorization-domain","display_name":"Unique factorization domain","score":0.6322051882743835},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.6139765977859497},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.5586221218109131},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5585313439369202},{"id":"https://openalex.org/keywords/rational-number","display_name":"Rational number","score":0.5031303763389587},{"id":"https://openalex.org/keywords/multinomial-distribution","display_name":"Multinomial distribution","score":0.495853990316391},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.49539053440093994},{"id":"https://openalex.org/keywords/unit","display_name":"Unit (ring theory)","score":0.4773676097393036},{"id":"https://openalex.org/keywords/unit-disk","display_name":"Unit disk","score":0.4551744759082794},{"id":"https://openalex.org/keywords/symbolic-computation","display_name":"Symbolic computation","score":0.4481672942638397},{"id":"https://openalex.org/keywords/factor","display_name":"Factor (programming language)","score":0.4159725308418274},{"id":"https://openalex.org/keywords/unit-circle","display_name":"Unit circle","score":0.41262954473495483},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.39964210987091064},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3948964774608612},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.3619019687175751},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2838762402534485},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.27660199999809265},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2587655186653137},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.22559019923210144},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.0951869785785675},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08666479587554932}],"concepts":[{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.8181967735290527},{"id":"https://openalex.org/C185592164","wikidata":"https://www.wikidata.org/wiki/Q1052579","display_name":"Unique factorization domain","level":3,"score":0.6322051882743835},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.6139765977859497},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.5586221218109131},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5585313439369202},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.5031303763389587},{"id":"https://openalex.org/C192065140","wikidata":"https://www.wikidata.org/wiki/Q1147928","display_name":"Multinomial distribution","level":2,"score":0.495853990316391},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.49539053440093994},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.4773676097393036},{"id":"https://openalex.org/C47177299","wikidata":"https://www.wikidata.org/wiki/Q2390323","display_name":"Unit disk","level":2,"score":0.4551744759082794},{"id":"https://openalex.org/C110812573","wikidata":"https://www.wikidata.org/wiki/Q175515","display_name":"Symbolic computation","level":2,"score":0.4481672942638397},{"id":"https://openalex.org/C2781039887","wikidata":"https://www.wikidata.org/wiki/Q1391724","display_name":"Factor (programming language)","level":2,"score":0.4159725308418274},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.41262954473495483},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.39964210987091064},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3948964774608612},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3619019687175751},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2838762402534485},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.27660199999809265},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2587655186653137},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.22559019923210144},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0951869785785675},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08666479587554932},{"id":"https://openalex.org/C19165224","wikidata":"https://www.wikidata.org/wiki/Q23404","display_name":"Anthropology","level":1,"score":0.0},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.0},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","level":1,"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},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1823931.1823934","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1823931.1823934","pdf_url":null,"source":{"id":"https://openalex.org/S176185422","display_name":"ACM communications in computer algebra","issn_l":"1932-2232","issn":["1932-2232","1932-2240"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320725","host_organization_name":"SIGSAM","host_organization_lineage":["https://openalex.org/P4310320725"],"host_organization_lineage_names":["SIGSAM"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Communications in Computer Algebra","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":1,"referenced_works":["https://openalex.org/W2079804015"],"related_works":["https://openalex.org/W4288313061","https://openalex.org/W2952928706","https://openalex.org/W29633042","https://openalex.org/W2056801132","https://openalex.org/W1982796873","https://openalex.org/W4287124158","https://openalex.org/W3171266272","https://openalex.org/W3213160425","https://openalex.org/W2043078341","https://openalex.org/W4386518712"],"abstract_inverted_index":{"Polynomials":[0],"over":[1,140,154],"the":[2,14,30,51,83,97,101,106,115,141,146,155,176,189],"Gaussian":[3,156],"integers":[4,142],"Z[":[5,17],"i":[6,18,23,26,120,123,127,132],"]":[7,19],"are":[8,20,109,166,183],"a":[9],"unique":[10],"factorization":[11],"domain,":[12],"and":[13,24,69,103,172],"units":[15,41],"in":[16,100,188],"1,":[21],"-1,":[22],"--":[25,119,131],".":[27],"To":[28],"implement":[29],"advantageous":[31],"uniqueness,":[32],"we":[33],"might":[34],"have":[35],"to":[36,47,60,72,117,150,168],"factor":[37],"one":[38],"of":[39,43,50,86,105,178,192],"these":[40],"out":[42],"some":[44],"multinomial":[45],"factors":[46,52,99],"make":[48],"all":[49],"unit":[53,152],"normal.":[54],"Not":[55],"doing":[56],"so":[57],"can":[58],"lead":[59],"unnecessarily":[61],"complicated":[62],"results,":[63],"disastrous":[64],"unrecognized":[65],"division":[66],"by":[67],"0,":[68],"lost":[70],"opportunities":[71],"obtain":[73],"closed-form":[74],"integrals,":[75],"sums,":[76],"limits,":[77],"equation":[78],"solutions,":[79],"etc.":[80],"For":[81],"example,":[82],"default":[84,190],"simplification":[85,191],"at":[87],"least":[88],"two":[89,162],"major":[90],"computer":[91,194],"algebra":[92,195],"systems":[93],"don't":[94],"recognize":[95],"that":[96],"linear":[98],"numerator":[102],"denominator":[104],"following":[107],"ratio":[108,116],"proportional,":[110],"hence":[111],"do":[112],"not":[113],"simplify":[114],"1:":[118],"((1":[121],"+":[122,126,135],")":[124,128,133],"z":[125,134],"/":[129],"(1":[130],"1":[136],"\u21921.":[137],"Unlike":[138],"polynomials":[139],"or":[143],"rational":[144],"numbers,":[145],"literature":[147],"doesn't":[148],"appear":[149],"address":[151],"normalization":[153],"integers.":[157],"Therefore":[158],"this":[159],"note":[160],"proposes":[161],"alternative":[163],"definitions.":[164],"They":[165],"easy":[167],"implement,":[169],"execute":[170],"quickly,":[171],"can't":[173],"greatly":[174],"increase":[175],"bulk":[177],"an":[179],"expression.":[180],"Consequently":[181],"they":[182],"good":[184],"candidates":[185],"for":[186],"inclusion":[187],"every":[193],"system.":[196]},"counts_by_year":[{"year":2014,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
