Factoring Rational Polynomials over the Complex Numbers
NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean ci...
Saved in:
| Published in: | SIAM journal on computing Vol. 22; no. 2; pp. 318 - 331 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Philadelphia, PA
Society for Industrial and Applied Mathematics
01.04.1993
|
| Subjects: | |
| ISSN: | 0097-5397, 1095-7111 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polynomial are its degree, coefficient length, number of variables ($d$, $c$, and $n$, respectively). If $n$ is fixed, we give a deterministic NC algorithm. If the number of variables is not fixed, we give a random (Monte-Carlo) NC algorithm in these input measures to find the number and degree of each irreducible factor. After reducing to the two-variable, square-free case, we apply the classical algebraic geometry fact that the absolute irreducible factors of $(P(z_1 ,z_2 ) = 0)$ correspond to the connected components of the real surface (or complex curve) $P(z_1 ,z_2 ) = 0$ minus its singular points. In finding the number of connected components of the surface $P = 0$, the surface is projected to the the $z_2 $-plane. The singular points of $P(z_1 ,z_2 )$ lie over the projection's critical values. The inverse image of a grid isolating the critical values in the $z_2 $-plane lifts to a one-dimensional real curve skeleton on the surface $(P = 0)$ whose number of connected components is precisely the number of connected components of $P = 0$ minus its singular points. The connectivity of this curve skeleton is constructed symbolically using Sturm sequences associated with the various polynomials defining these maps. Given the number of irreducible factors and their degrees, the actual factors can be reconstructed using the recent result of Neff [Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pp. 152-162] on finding zeros of one-variable polynomials in NC. |
|---|---|
| AbstractList | NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polynomial are its degree, coefficient length, number of variables ($d$, $c$, and $n$, respectively). If $n$ is fixed, we give a deterministic NC algorithm. If the number of variables is not fixed, we give a random (Monte-Carlo) NC algorithm in these input measures to find the number and degree of each irreducible factor. After reducing to the two-variable, square-free case, we apply the classical algebraic geometry fact that the absolute irreducible factors of $(P(z_1 ,z_2 ) = 0)$ correspond to the connected components of the real surface (or complex curve) $P(z_1 ,z_2 ) = 0$ minus its singular points. In finding the number of connected components of the surface $P = 0$, the surface is projected to the the $z_2 $-plane. The singular points of $P(z_1 ,z_2 )$ lie over the projection's critical values. The inverse image of a grid isolating the critical values in the $z_2 $-plane lifts to a one-dimensional real curve skeleton on the surface $(P = 0)$ whose number of connected components is precisely the number of connected components of $P = 0$ minus its singular points. The connectivity of this curve skeleton is constructed symbolically using Sturm sequences associated with the various polynomials defining these maps. Given the number of irreducible factors and their degrees, the actual factors can be reconstructed using the recent result of Neff [Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pp. 152-162] on finding zeros of one-variable polynomials in NC. |
| Author | Garrity, Thomas Warren, Joe Bajaj, Chanderjit Canny, John |
| Author_xml | – sequence: 1 givenname: Chanderjit surname: Bajaj fullname: Bajaj, Chanderjit – sequence: 2 givenname: John surname: Canny fullname: Canny, John – sequence: 3 givenname: Thomas surname: Garrity fullname: Garrity, Thomas – sequence: 4 givenname: Joe surname: Warren fullname: Warren, Joe |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=4236440$$DView record in Pascal Francis |
| BookMark | eNpl0E1LAzEQgOEgFWyr-BcWETyt5nOTHKVYFYqK6HlJsomm7G5qkkr7711t8aCnOczDMLwTMOpDbwE4RfASIcKvIMYYYnoAxghKVnKE0AiMIZS8ZETyIzBJaQkhohSRMeBzZXKIvn8rnlX2oVdt8RTabR86r9pUhE8bi_xui1noVq3dFA_rTtuYjsGhG_b2ZD-n4HV-8zK7KxePt_ez60VpMJe5RMIxKxqGsRXOaKolI8I5LRtlKBUQV0JL20jHhIaCMiNIozhstHKkIlKRKTjb3V3F8LG2KdfLsI7Dl6mWSFYSUykHdL5HKhnVuqh641O9ir5TcVtTTCpK4cAudszEkFK07lcgWH-3q_ftBln-kcbnnzw5Kt_-818-iXAH |
| CitedBy_id | crossref_primary_10_1006_jsco_2002_0564 crossref_primary_10_1016_j_tcs_2010_11_047 crossref_primary_10_1006_jcss_2000_1728 crossref_primary_10_1016_j_jco_2007_01_008 crossref_primary_10_1016_j_jsc_2010_06_020 crossref_primary_10_1016_j_tcs_2004_01_011 crossref_primary_10_1016_j_jsc_2007_01_003 crossref_primary_10_1006_jsco_2001_0461 crossref_primary_10_1090_S0025_5718_02_01428_X |
| Cites_doi | 10.1016/S0019-9958(85)80056-5 10.1007/3-540-10843-2_2 10.1017/9781139171984 10.1007/BF01457454 10.1145/322217.322225 10.1007/BFb0032834 10.1007/BF01449599 10.1137/0214035 10.1109/FSCS.1990.89534 10.1016/0022-0000(85)90043-1 10.1016/S0019-9958(82)90766-5 10.1007/3-540-51082-6_79 10.1007/978-1-4615-6899-5 10.1016/S0747-7171(85)80029-8 10.1007/978-1-4757-3849-0 10.1137/0212043 10.1007/978-3-642-96200-4 |
| ContentType | Journal Article |
| Copyright | 1994 INIST-CNRS [Copyright] © 1993 © Society for Industrial and Applied Mathematics |
| Copyright_xml | – notice: 1994 INIST-CNRS – notice: [Copyright] © 1993 © Society for Industrial and Applied Mathematics |
| DBID | AAYXX CITATION IQODW 3V. 7RQ 7WY 7WZ 7X2 7XB 87Z 88A 88F 88I 88K 8AL 8FE 8FG 8FH 8FK 8FL 8G5 ABJCF ABUWG AFKRA ARAPS ATCPS AZQEC BBNVY BENPR BEZIV BGLVJ BHPHI CCPQU D1I DWQXO FRNLG F~G GNUQQ GUQSH HCIFZ JQ2 K60 K6~ K7- KB. L.- L6V LK8 M0C M0K M0N M1Q M2O M2P M2T M7P M7S MBDVC P5Z P62 PATMY PDBOC PHGZM PHGZT PKEHL PQBIZ PQBZA PQEST PQGLB PQQKQ PQUKI PRINS PTHSS PYCSY Q9U S0W U9A |
| DOI | 10.1137/0222024 |
| DatabaseName | CrossRef Pascal-Francis ProQuest Central (Corporate) Career & Technical Education Database ABI/INFORM Collection ABI/INFORM Global (PDF only) Agricultural Science Collection ProQuest Central (purchase pre-March 2016) ABI/INFORM Collection Biology Database (Alumni Edition) Military Database (Alumni Edition) Science Database (Alumni Edition) Telecommunications (Alumni Edition) Computing Database (Alumni Edition) ProQuest SciTech Collection ProQuest Technology Collection ProQuest Natural Science Collection ProQuest Central (Alumni) (purchase pre-March 2016) ABI/INFORM Collection (Alumni) ProQuest Research Library Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection Agricultural & Environmental Science Collection ProQuest Central Essentials Biological Science Collection ProQuest Central Business Premium Collection Technology Collection Natural Science Collection ProQuest One ProQuest Materials Science Collection ProQuest Central Korea Business Premium Collection (Alumni) ABI/INFORM Global (Corporate) ProQuest Central Student ProQuest Research Library SciTech Premium Collection ProQuest Computer Science Collection ProQuest Business Collection (Alumni Edition) ProQuest Business Collection Computer Science Database (ProQuest) Materials Science Database ABI/INFORM Professional Advanced ProQuest Engineering Collection Biological Sciences ABI/INFORM Global Agricultural Science Database Computing Database Military Database Research Library Science Database (ProQuest) Telecommunications Database Biological Science Database Engineering Database Research Library (Corporate) Advanced Technologies & Aerospace Database ProQuest Advanced Technologies & Aerospace Collection Environmental Science Database Materials Science Collection Proquest Central Premium ProQuest One Academic (New) ProQuest One Academic Middle East (New) ProQuest One Business ProQuest One Business (Alumni) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection Environmental Science Collection ProQuest Central Basic DELNET Engineering & Technology Collection |
| DatabaseTitle | CrossRef Agricultural Science Database ProQuest Business Collection (Alumni Edition) Research Library Prep Computer Science Database ProQuest Central Student ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection SciTech Premium Collection ProQuest Military Collection ProQuest Central China ABI/INFORM Complete ProQuest Telecommunications ProQuest One Applied & Life Sciences Natural Science Collection Biological Science Collection ProQuest Central (New) Engineering Collection Career and Technical Education (Alumni Edition) Advanced Technologies & Aerospace Collection Business Premium Collection ABI/INFORM Global Engineering Database ProQuest Science Journals (Alumni Edition) ProQuest Biological Science Collection ProQuest One Academic Eastern Edition Agricultural Science Collection ProQuest Technology Collection ProQuest Telecommunications (Alumni Edition) Biological Science Database ProQuest Business Collection Environmental Science Collection ProQuest Career and Technical Education ProQuest One Academic UKI Edition Environmental Science Database ProQuest One Academic ProQuest One Academic (New) ABI/INFORM Global (Corporate) ProQuest One Business Technology Collection ProQuest One Academic Middle East (New) Materials Science Collection ProQuest Central (Alumni Edition) ProQuest One Community College Research Library (Alumni Edition) ProQuest Natural Science Collection ProQuest Biology Journals (Alumni Edition) ProQuest Central ABI/INFORM Professional Advanced ProQuest Engineering Collection ProQuest Central Korea Agricultural & Environmental Science Collection Materials Science Database ProQuest Research Library ABI/INFORM Complete (Alumni Edition) ProQuest Materials Science Collection ProQuest Computing ABI/INFORM Global (Alumni Edition) ProQuest Central Basic ProQuest Science Journals ProQuest Computing (Alumni Edition) ProQuest Military Collection (Alumni Edition) ProQuest SciTech Collection Advanced Technologies & Aerospace Database ProQuest DELNET Engineering and Technology Collection Materials Science & Engineering Collection ProQuest One Business (Alumni) ProQuest Central (Alumni) Business Premium Collection (Alumni) |
| DatabaseTitleList | Agricultural Science Database |
| Database_xml | – sequence: 1 dbid: BENPR name: ProQuest Central url: https://www.proquest.com/central sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics Computer Science |
| EISSN | 1095-7111 |
| EndPage | 331 |
| ExternalDocumentID | 2577571401 4236440 10_1137_0222024 |
| GroupedDBID | --Z -DZ -~X .4S .DC 123 4.4 7RQ 7WY 7X2 7XC 88I 8CJ 8FE 8FG 8FH 8FL 8G5 8V8 AALVN AASXH AAYXX ABDBF ABDPE ABJCF ABKAD ABMZU ABPPZ ABUFD ABUWG ACBEA ACGFO ACGOD ACIWK ACNCT ACPRK ACUHS ADBBV ADMHC ADXHL AEMOZ AENEX AFFHD AFKRA AFRAH AHQJS AKVCP ALMA_UNASSIGNED_HOLDINGS ANXRF ARAPS ARCSS ATCPS AZQEC BBNVY BENPR BEZIV BGLVJ BHPHI BPHCQ CCPQU CITATION CS3 CZ9 D1I D1J D1K DQ2 DU5 DWQXO EAP EAS EBR EBS EBU ECS EDO EJD EMK EST ESX FRNLG GNUQQ GUQSH H13 HCIFZ H~9 I-F K1G K6- K60 K6V K6~ K7- KB. KC. L6V LK5 LK8 M0C M0K M1Q M2O M2P M7P M7R M7S MK~ NHB OHT P1Q P2P P62 PATMY PDBOC PHGZM PHGZT PQBIZ PQBZA PQGLB PQQKQ PROAC PTHSS PYCSY RJG RNS RSI S0W TAE TH9 TN5 TUS TWZ UHB UQL VOH WH7 WHG XJE YNT YYP YZZ ZCG ZY4 IQODW XJT 3V. 7XB 88A 88K 8AL 8FK JQ2 L.- M0N M2T MBDVC PKEHL PQEST PQUKI PRINS Q9U U9A |
| ID | FETCH-LOGICAL-c279t-18f5e8d522e8fcb4b9538ffb9dac4480268b9ed9f58b0845c83da70dbaf3639a3 |
| IEDL.DBID | M7P |
| ISICitedReferencesCount | 12 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=10_1137_0222024&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0097-5397 |
| IngestDate | Fri Jul 25 11:10:42 EDT 2025 Mon Jul 21 09:13:15 EDT 2025 Sat Nov 29 03:54:32 EST 2025 Tue Nov 18 21:39:35 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 2 |
| Keywords | Polynomial Critical value N variable function Singular point Factorization Algebraic geometry Complex number |
| Language | English |
| License | CC BY 4.0 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c279t-18f5e8d522e8fcb4b9538ffb9dac4480268b9ed9f58b0845c83da70dbaf3639a3 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14 |
| PQID | 919692499 |
| PQPubID | 666313 |
| PageCount | 14 |
| ParticipantIDs | proquest_journals_919692499 pascalfrancis_primary_4236440 crossref_primary_10_1137_0222024 crossref_citationtrail_10_1137_0222024 |
| PublicationCentury | 1900 |
| PublicationDate | 1993-04-01 |
| PublicationDateYYYYMMDD | 1993-04-01 |
| PublicationDate_xml | – month: 04 year: 1993 text: 1993-04-01 day: 01 |
| PublicationDecade | 1990 |
| PublicationPlace | Philadelphia, PA |
| PublicationPlace_xml | – name: Philadelphia, PA – name: Philadelphia |
| PublicationTitle | SIAM journal on computing |
| PublicationYear | 1993 |
| Publisher | Society for Industrial and Applied Mathematics |
| Publisher_xml | – name: Society for Industrial and Applied Mathematics |
| References | R7 R9 R10 R20 R23 R22 R25 R13 R16 Mumford D. (R21) 1970 R27 R15 R26 Hartshorne Robin (R12) 1977 R17 Henrici Peter (R14) 1988 R19 Griffiths Phillip (R11) 1978 R1 Ciarlet P. (R5) 1989 |
| References_xml | – ident: R17 doi: 10.1016/S0019-9958(85)80056-5 – ident: R13 doi: 10.1007/3-540-10843-2_2 – volume-title: Introduction to numerical linear algebra and optimisation year: 1989 ident: R5 doi: 10.1017/9781139171984 – ident: R20 doi: 10.1007/BF01457454 – ident: R25 doi: 10.1145/322217.322225 – ident: R7 doi: 10.1007/BFb0032834 – ident: R23 doi: 10.1007/BF01449599 – volume-title: Applied and computational complex analysis. Vol. 1 year: 1988 ident: R14 – ident: R16 doi: 10.1137/0214035 – ident: R22 doi: 10.1109/FSCS.1990.89534 – volume-title: Algebraic Geometry I: Complex Projective Varieties year: 1970 ident: R21 – volume-title: Principles of algebraic geometry year: 1978 ident: R11 – ident: R10 doi: 10.1016/0022-0000(85)90043-1 – ident: R1 doi: 10.1016/S0019-9958(82)90766-5 – ident: R9 doi: 10.1007/3-540-51082-6_79 – ident: R19 doi: 10.1007/978-1-4615-6899-5 – ident: R15 doi: 10.1016/S0747-7171(85)80029-8 – volume-title: Algebraic geometry year: 1977 ident: R12 doi: 10.1007/978-1-4757-3849-0 – ident: R27 doi: 10.1137/0212043 – ident: R26 doi: 10.1007/978-3-642-96200-4 |
| SSID | ssj0014413 |
| Score | 1.4681574 |
| Snippet | NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial... |
| SourceID | proquest pascalfrancis crossref |
| SourceType | Aggregation Database Index Database Enrichment Source |
| StartPage | 318 |
| SubjectTerms | Algebra Algorithms Computer science Exact sciences and technology Field theory and polynomials Geometry Mathematics Polynomials Sciences and techniques of general use Variables |
| Title | Factoring Rational Polynomials over the Complex Numbers |
| URI | https://www.proquest.com/docview/919692499 |
| Volume | 22 |
| WOSCitedRecordID | wos10_1137_0222024&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVPQU databaseName: ABI/INFORM Collection customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: 7WY dateStart: 19720301 isFulltext: true titleUrlDefault: https://www.proquest.com/abicomplete providerName: ProQuest – providerCode: PRVPQU databaseName: ABI/INFORM Global customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: M0C dateStart: 19720301 isFulltext: true titleUrlDefault: https://search.proquest.com/abiglobal providerName: ProQuest – providerCode: PRVPQU databaseName: Advanced Technologies & Aerospace Database customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: P5Z dateStart: 19720301 isFulltext: true titleUrlDefault: https://search.proquest.com/hightechjournals providerName: ProQuest – providerCode: PRVPQU databaseName: Agricultural Science Database customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: M0K dateStart: 19720301 isFulltext: true titleUrlDefault: https://search.proquest.com/agriculturejournals providerName: ProQuest – providerCode: PRVPQU databaseName: Biological Science Database customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: M7P dateStart: 19720301 isFulltext: true titleUrlDefault: http://search.proquest.com/biologicalscijournals providerName: ProQuest – providerCode: PRVPQU databaseName: Computer Science Database customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: K7- dateStart: 19720301 isFulltext: true titleUrlDefault: http://search.proquest.com/compscijour providerName: ProQuest – providerCode: PRVPQU databaseName: Engineering Database customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: M7S dateStart: 19720301 isFulltext: true titleUrlDefault: http://search.proquest.com providerName: ProQuest – providerCode: PRVPQU databaseName: Environmental Science Database customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: PATMY dateStart: 19720301 isFulltext: true titleUrlDefault: http://search.proquest.com/environmentalscience providerName: ProQuest – providerCode: PRVPQU databaseName: Materials Science Database customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: KB. dateStart: 19720301 isFulltext: true titleUrlDefault: http://search.proquest.com/materialsscijournals providerName: ProQuest – providerCode: PRVPQU databaseName: Military Database (ProQuest) customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: M1Q dateStart: 19720301 isFulltext: true titleUrlDefault: https://search.proquest.com/military providerName: ProQuest – providerCode: PRVPQU databaseName: Proquest Career & Technical Education customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: 7RQ dateStart: 19720301 isFulltext: true titleUrlDefault: https://search.proquest.com/career providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest Central customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: BENPR dateStart: 19720301 isFulltext: true titleUrlDefault: https://www.proquest.com/central providerName: ProQuest – providerCode: PRVPQU databaseName: Research Library customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: M2O dateStart: 19720301 isFulltext: true titleUrlDefault: https://search.proquest.com/pqrl providerName: ProQuest – providerCode: PRVPQU databaseName: Science Database customDbUrl: eissn: 1095-7111 dateEnd: 20140531 omitProxy: false ssIdentifier: ssj0014413 issn: 0097-5397 databaseCode: M2P dateStart: 19720301 isFulltext: true titleUrlDefault: https://search.proquest.com/sciencejournals providerName: ProQuest |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3db9MwED-xjQcQ2mAM0X1UfkC8GZI4qe2niU2rkEZLKAMGL1Hs2BJS1ZalIPjvuXPcSNMkXnj5SVFOUeSz78M-_w7gBTGUmMYbrnTqeJ7KmqPTEzx1hcgaXdssUAp9fienU3V9rctYm9PGssqNTQyGulla2iN_rYnHBXMFfbr6walpFB2uxg4aW7BDJAkiVO6V_SECevpIQil5gX63uzObpkTpg24xyfJbzujRqm5xXHzX0OKObQ4OZ7z3n7_6GHZjpMnedFPjCdxzi33Y23RxYHFR78PDSc_c2j4FOQ4NeNChsVncJ2Tlcv6Hbi_jTGVU8clQntGH5u43m4aWIu0BfBpfXJ2_5bG5AqfhX_NU-cKpBsMvp7w1udFo-rw3uqktpmyYmimjXaN9oUyi8sIq0dQyaUztBUY1tXgG24vlwj0HNrLWOCWtpPivaEZqlBjnhfBWY_6ZyQG83IxyZSPzODXAmFchAxGyiuoYAOsFVx3Zxl2Rk1tq6uVyYsLPkwEcbXRSxdXYVr1CDv_59ggedOWMVJdzDNvrm5_uBO7bX-vv7c0QtuTsA-GXr0PYObuYljN8upSc8OwV4iQ5D3hJmJLsJHsfsAx4NQzzM-BHxLL49heNveP_ |
| linkProvider | ProQuest |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1bSxtBFD7YWKgi2qYWo6adh7ZvQ_eamXkQETUoJouUtORtu3MDISQxm7b6o_yPntkbSMA3H3zewyy758z5zpnL9wF8dQwlUltJufANjXyWUQS9kPomDgMtMhUUlEK_ByxJ-Hgsrtfgob4L445V1jmxSNR6ptwa-Q_heFywVxDH81vqRKPc5mqtoFFGxZW5_48dW350eYbu_RYE_fPR6QWtRAWoe-2S-tzGhmssOwy3SkZS4JS3VgqdKWxVsCXhUhgtbMylx6NY8VBnzNMysyGieRbiuG9gPcJGyE2oYTBqNi2wsqhILxmNEefLO7q-7yiEEIa9IHoCflvzLEc_2FJAYwULCoDr77yyX_MetqtKmpyUof8B1sy0DTu1SgWpklYbNocNM23-EVi_EBhCwCY_q3VQcj2b3Lvb2TgTiTvRStCeuIEm5o4khWRKvgu_XuRjPkFrOpuaPSA9paThTDFX38a6x3ueNDYMrRLYXwesA99rr6aqYlZ3Ah-TtOiwQpZW7u8AaQznJZnIqkn3SVg0dpFj-o-8DhzUMZBW2SZPmwDYf_bpF3h3MRoO0sFlcnUAG-XRTXcG6RBay8Vf04W36t_yJl98LuKawJ-XjpZH3NE1_w |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Factoring+Rational+Polynomials+over+the+Complex+Numbers&rft.jtitle=SIAM+journal+on+computing&rft.au=Bajaj%2C+Chanderjit&rft.au=Canny%2C+John&rft.au=Garrity%2C+Thomas&rft.au=Warren%2C+Joe&rft.date=1993-04-01&rft.issn=0097-5397&rft.eissn=1095-7111&rft.volume=22&rft.issue=2&rft.spage=318&rft.epage=331&rft_id=info:doi/10.1137%2F0222024&rft.externalDBID=n%2Fa&rft.externalDocID=10_1137_0222024 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0097-5397&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0097-5397&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0097-5397&client=summon |