Modern Computer Algebra

812 Pages·2013·2.05 MB·English

by  Joachim von zur Gathen

Modern Computer Algebra
Computeralgebrasystemsarenowubiquitousinallareasofscienceandengineer-
ing. This highly successful textbook, widely regarded as the “bible of computer
algebra”,givesathoroughintroductiontothealgorithmicbasisofthemathematical
engineincomputeralgebrasystems.Designedtoaccompanyone-ortwo-semester
courses for advanced undergraduate or graduate students in computer science or
mathematics, its comprehensiveness and reliability has also made it an essential
referenceforprofessionalsinthearea.
Special features include: detailed study of algorithms including time analysis;
implementation reports on several topics; complete proofs of the mathematical
underpinnings; and a wide variety of applications (among others, in chemistry,
codingtheory,cryptography,computationallogic,andthedesignofcalendarsand
musicalscales).Agreatdealofhistoricalinformationandillustrationenlivensthe
text.
Inthisthirdedition,errorshavebeencorrectedandmuchoftheFastEuclidean
Algorithmchapterhasbeenrenovated.
JoachimvonzurGathenhasaPhDfromUniversitätZürichandhastaughtatthe
UniversityofTorontoandtheUniversityofPaderborn.Heiscurrentlyaprofessor
at the Bonn–Aachen International Center for Information Technology (B-IT) and
theDepartmentofComputerScienceatUniversitätBonn.
Jürgen Gerhard has a PhD from Universität Paderborn. He is now Director of
Research at Maplesoft in Canada, where he leads research collaborations with
partners in Canada, France, Russia, Germany, the USA, and the UK, as well as
anumberofconsultingprojectsforglobalplayersintheautomotiveindustry.

Modern Computer Algebra
Third Edition
JOACHIM VON ZUR GATHEN
Bonn–AachenInternationalCenter
forInformationTechnology(B-IT)
JÜRGEN GERHARD
Maplesoft,Waterloo
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107039032
Firstandsecondeditions(cid:2)c CambridgeUniversityPress1999,2003
Thirdedition(cid:2)c JoachimvonzurGathenandJürgenGerhard2013
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished1999
Secondedition2003
Thirdedition2013.
PrintedandboundbyiCPIiGroupi(UK)iLtd,iCroydoniCR0i4i4YY
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-1-107-03903-2Hardback
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ToDorothea,Rafaela,Désirée
Forendlesspatience
ToMercedesCappuccino

Contents
Introduction 1
1 Cyclohexane,cryptography,codes,andcomputeralgebra 11
1.1 Cyclohexaneconformations . . . . . . . . . . . . . . . . . . . . 11
1.2 TheRSAcryptosystem . . . . . . . . . . . . . . . . . . . . . . 16
1.3 Distributeddatastructures . . . . . . . . . . . . . . . . . . . . . 18
1.4 Computeralgebrasystems . . . . . . . . . . . . . . . . . . . . . 19
I Euclid 23
2 Fundamentalalgorithms 29
2.1 Representationandadditionofnumbers . . . . . . . . . . . . . . 29
2.2 Representationandadditionofpolynomials . . . . . . . . . . . . 32
2.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Divisionwithremainder . . . . . . . . . . . . . . . . . . . . . . 37
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 TheEuclideanAlgorithm 45
3.1 Euclideandomains . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 TheExtendedEuclideanAlgorithm . . . . . . . . . . . . . . . . 47
3.3 CostanalysisforZandF[x] . . . . . . . . . . . . . . . . . . . . 51
3.4 (Non-)Uniquenessofthegcd . . . . . . . . . . . . . . . . . . . 55
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 ApplicationsoftheEuclideanAlgorithm 69
4.1 Modulararithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 ModularinversesviaEuclid . . . . . . . . . . . . . . . . . . . . 73
4.3 Repeatedsquaring . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 ModularinversesviaFermat . . . . . . . . . . . . . . . . . . . . 76
vii
viii Contents
4.5 LinearDiophantineequations . . . . . . . . . . . . . . . . . . . 77
4.6 ContinuedfractionsandDiophantineapproximation . . . . . . . 79
4.7 Calendars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.8 Musicalscales . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Modularalgorithmsandinterpolation 97
5.1 Changeofrepresentation . . . . . . . . . . . . . . . . . . . . . 100
5.2 Evaluationandinterpolation . . . . . . . . . . . . . . . . . . . . 101
5.3 Application: Secretsharing . . . . . . . . . . . . . . . . . . . . 103
5.4 TheChineseRemainderAlgorithm . . . . . . . . . . . . . . . . 104
5.5 Modulardeterminantcomputation . . . . . . . . . . . . . . . . . 109
5.6 Hermiteinterpolation . . . . . . . . . . . . . . . . . . . . . . . 113
5.7 Rationalfunctionreconstruction . . . . . . . . . . . . . . . . . . 115
5.8 Cauchyinterpolation . . . . . . . . . . . . . . . . . . . . . . . . 118
5.9 Padéapproximation . . . . . . . . . . . . . . . . . . . . . . . . 121
5.10 Rationalnumberreconstruction . . . . . . . . . . . . . . . . . . 124
5.11 Partialfractiondecomposition . . . . . . . . . . . . . . . . . . . 128
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Theresultantandgcdcomputation 141
6.1 CoefficientgrowthintheEuclideanAlgorithm . . . . . . . . . . 141
6.2 Gauß’lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3 Theresultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.4 Modulargcdalgorithms . . . . . . . . . . . . . . . . . . . . . . 158
6.5 ModulargcdalgorithminF[x,y] . . . . . . . . . . . . . . . . . 161
6.6 Mignotte’sfactorboundandamodulargcdalgorithminZ[x] . . 164
6.7 Smallprimesmodulargcdalgorithms . . . . . . . . . . . . . . . 168
6.8 Application: intersectingplanecurves . . . . . . . . . . . . . . . 171
6.9 Nonzeropreservationandthegcdofseveralpolynomials. . . . . 176
6.10 Subresultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.11 ModularExtendedEuclideanAlgorithms . . . . . . . . . . . . . 183
6.12 PseudodivisionandprimitiveEuclideanAlgorithms . . . . . . . 190
6.13 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7 Application: DecodingBCHcodes 209
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215