Table Of Content185
Graduate Texts in Mathematics
EditorialBoard
S.Axler F.W.Gehring K.A.Ribet
David A. Cox
John Little
Donal O’Shea
Using Algebraic Geometry
Second Edition
With24Illustrations
DavidCox DonalO’Shea
DepartmentofMathematics DepartmentofMathematics
AmherstCollege MountHolyokeCollege
Amherst,MA01002-5000 SouthHadley,MA01075
USA USA
[email protected] [email protected]
JohnLittle
DepartmentofMathematics
CollegeoftheHolyCross
Worcester,MA01610
USA
[email protected]
EditorialBoard
S.Axler F.W.Gehring K.A.Ribet
MathematicsDepartment MathematicsDepartment DepartmentofMathematics
SanFranciscoState EastHall UniversityofCalifornia,
University UniversityofMichigan Berkeley
SanFrancisco,CA94132 AnnArbor,MI48109 Berkeley,CA94720-3840
USA USA USA
MathematicsSubjectClassification(2000):13Pxx,13-01,14-01,14Qxx
LibraryofCongressCataloging-in-PublicationData
Little,JohnB.
Usingalgebraicgeometry/JohnLittle,DavidA.Cox,DonalO’Shea.
p. cm.—(Graduatetextsinmathematics;v.185)
Cox’snameappearsfirstontheearlieredition.
Includesbibliographicalreferencesandindex.
ISBN0-387-20706-6(alk.paper)–ISBN0-387-20733-3(pbk.:alk.paper)
1. Geometry,Algebraic. I. Cox,DavidA. II. O’Shea,Donal. III. Title. IV. Graduate
textsinmathematics;185.
QA564.C6883 2004
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516.35—dc22 2003070363
ISBN0-387-20706-6(hardcover) Printedonacid-freepaper.
ISBN0-387-20733-3(softcover)
(cid:2)c 2005,1998SpringerScience+BusinessMedia,Inc.
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Preface to the Second Edition
Since the first edition of Using Algebraic Geometry was published in 1998, the
fieldofcomputationalalgebraicgeometryanditsapplicationshasdevelopedrap-
idly.Manynewresultsconcerningtopicswediscussedhaveappeared.Moreover,
anumberofnewintroductorytextshavebeenpublished.Ourgoalsinthisrevision
have been to update the references to reflect these additions to the literature, to
adddiscussionsofsomenewmaterial,toimprovesomeoftheproofs,andtofix
typographicalerrors. Themajorchangesinthiseditionarethefollowing:
• Aunifieddiscussionofhowmatricescanbeusedtospecifymonomialor-
dersin§2ofChapter1.
• ArewrittenpresentationoftheMoranormalformalgorithmin§3ofChap-
ter4andthedivisionof§4intotwosections.
• TheadditionoftwosectionsinChapter8:§4introducestheGro¨bnerfanof
anidealand§5discussestheGro¨bnerWalkbasisconversionalgorithm.
• The replacement of §5 of Chapter 9 by a new Chapter 10 on the theory
oforderdomains,associatedcodes,andtheBerlekamp-Massey-Sakatade-
codingalgorithm.Theone-pointgeometricGoppacodesstudiedinthefirst
editionarespecialcasesofthisconstruction.
• The Maple code has been updated and Macaulay has been replaced by
Macaulay2.
We would like to thank the many readers who helped us find typographical
errorsinthe first edition. Special thanks goto Rainer Steinwandtfor his heroic
efforts. We also want to give particular thanks to Rex Agacy, Alicia Dicken-
stein, Dan Grayson, Serkan Hos¸ten, Christoph Ko¨gl, Nick Loehr, Jim Madden,
MikeO’Sullivan,LyleRamshaw,HalSchenck,HansSterk,MikeStillman,Bernd
Sturmfels,andIrenaSwansonfortheirhelp.
August,2004 DavidCox
JohnLittle
DonalO’Shea
v
Preface to the First Edition
In recent years, the discovery of new algorithms for dealing with polynomial
equations,coupledwiththeirimplementationoninexpensiveyetfastcomputers,
has sparked a minor revolution in the study and practice of algebraic geometry.
Thesealgorithmicmethodsandtechniqueshavealsogivenrisetosomeexciting
newapplicationsofalgebraicgeometry.
OneofthegoalsofUsingAlgebraicGeometryistoillustratethemanyuses
of algebraic geometry and to highlight the more recent applications of Gro¨bner
basesandresultants. Inordertodothis,wealsoprovideanintroductiontosome
algebraicobjectsandtechniquesmoreadvancedthanonetypicallyencountersin
a first course, but which are nonetheless of great utility. Finally, we wanted to
write a book which would be accessible to nonspecialists and to readers with a
diverserangeofbackgrounds.
Tokeepthebookreasonablyshort, weoftenhavetorefertobasicresultsin
algebraic geometry without proof, although complete references are given. For
readers learning algebraic geometry and Gro¨bner bases for the first time, we
would recommend that they read this book in conjunction with one of the fol-
lowingintroductionstothesesubjects:
• IntroductiontoGro¨bnerBases,byAdamsandLoustaunau[AL]
• Gro¨bnerBases,byBeckerandWeispfenning[BW]
• Ideals,VarietiesandAlgorithms,byCox,LittleandO’Shea[CLO]
We have tried, on the other hand, to keep the exposition self-contained outside
of references to these introductory texts. We have made no effort at complete-
ness,andhavenothesitatedtopointthereadertotheresearchliteratureformore
information.
Laterintheprefacewewillgiveabriefsummaryofwhatourbookcovers.
TheLevelof the Text
Thisbookiswrittenatthegraduatelevelandhenceassumesthereaderknowsthe
material covered in standard undergraduate courses, including abstract algebra.
vii
viii PrefacetotheFirstEdition
But because the text is intended for beginning graduate students, it does not re-
quiregraduatealgebra,andinparticular,thebookdoesnotassumethatthereader
isfamiliarwithmodules. Beingagraduatetext,UsingAlgebraicGeometrycov-
ersmoresophisticatedtopicsandhasadenserexpositionthanmostundergraduate
texts,includingourpreviousbook[CLO].
However, itis possible touse this bookat theundergraduatelevel, provided
proper precautions are taken. With the exception of the first two chapters, we
foundthatmostundergraduatesneededhelpreadingpreliminaryversionsofthe
text. Thatsaid,ifonesupplementstheotherchapterswithsimplerexercisesand
fuller explanations, many of the applications we cover make good topics for an
upper-levelundergraduateappliedalgebracourse. Similarly,thebookcouldalso
beusedforreadingcoursesorseniorthesesatthislevel. Wehopethatourbook
will encourage instructors to find creative ways for involving advanced under-
graduatesinthiswonderfulmathematics.
Howto Usethe Text
Thebookcoversavarietyoftopics,whichcanbegroupedroughlyasfollows:
• Chapters 1 and 2: Gro¨bner bases, including basic definitions, algorithms
and theorems, together with solving equations, eigenvalue methods, and
solutionsoverR.
• Chapters3and7: Resultants,includingmultipolynomialandsparseresul-
tantsaswellastheirrelationtopolytopes,mixedvolumes, toricvarieties,
andsolvingequations.
• Chapters4,5and6: Commutativealgebra,includinglocalrings,standard
bases,modules,syzygies,freeresolutions,Hilbertfunctionsandgeometric
applications.
• Chapters8and9: Applications,includingintegerprogramming,combina-
torics,polynomialsplines,andalgebraiccodingtheory.
Oneunusualfeatureofthebook’sorganizationistheearlyintroductionofresul-
tants in Chapter 3. This is because there are many applications where resultant
methods are much more efficient than Gro¨bner basis methods. While Gro¨bner
basismethodshavehadagreatertheoreticalimpactonalgebraicgeometry,resul-
tantsappeartohaveanadvantagewhenitcomestopracticalapplications. There
isalsosomelovelymathematicsconnectedwithresultants.
There is a large degree of independence among most chapters of the book.
Thisimpliesthattherearemanywaysthebookcanbeusedinteachingacourse.
Sincethereismorematerialthancanbecoveredinonesemester, somechoices
arenecessary.Herearethreeexamplesofhowtostructureacourseusingourtext.
• Solving Equations. This course would focus on the use of Gro¨bner bases
andresultants tosolve systems of polynomialequations. Chapters1, 2, 3
PrefacetotheFirstEdition ix
and7wouldformtheheartofthecourse.Specialemphasiswouldbeplaced
on§5ofChapter2,§5and§6ofChapter3,and§6ofChapter7. Optional
topicswouldinclude§1and§2ofChapter4,whichdiscussmultiplicities.
• CommutativeAlgebra. Here, the focus would be on topics fromclassical
commutativealgebra. ThecoursewouldfollowChapters1, 2, 4, 5and6,
skipping only those parts of §2 of Chapter 4 which deal with resultants.
ThefinalsectionofChapter6isaniceendingpointforthecourse.
• Applications. A course concentrating on applications would cover inte-
gerprogramming,combinatorics,splinesandcodingtheory. Afteraquick
tripthroughChapters1and2,themainfocuswouldbeChapters8and9.
Chapter8usessomeideasaboutpolytopesfrom§1ofChapter7,andmod-
ules appear naturallyin Chapters8 and 9. Hence the first two sections of
Chapter 5 would need to be covered. Also, Chapters 8 and 9 use Hilbert
functions,whichcanbefoundineitherChapter6ofthisbookorChapter9
of[CLO].
Wewant to emphasizethat these areonlythreeof manyways of usingthe text.
We would be very interested in hearing from instructors who have found other
pathsthroughthebook.
References
Referencestothebibliographyattheendofthebookarebythefirstthreeletters
oftheauthor’slastname(e.g.,[Hil]forHilbert),withnumbersformultiplepapers
bythesameauthor(e.g.,[Mac1]forthefirstpaperbyMacaulay). Whenthereis
more than one author, the first letters of the authors’ last names are used (e.g.,
[AM]forAtiyahandMacdonald),andwhenseveralsetsofauthorshavethesame
initials,otherlettersareusedtodistinguishthem(e.g.,[BoF]isbyBonnesenand
Fenchel,while[BuF]isbyBurdenandFaires).
The bibliography lists books alphabetically by the full author’s name, fol-
lowed (if applicable) by any coauthors. This means, for instance, that [BS] by
BilleraandSturmfelsislistedbefore[Bla]byBlahut.
CommentsandCorrections
Weencouragecomments,criticism,andcorrections. Pleasesendthemtoanyof
us:
DavidCox [email protected]
JohnLittle [email protected]
DonO’Shea [email protected]
For eachnewtypoorerror, wewill pay$1tothefirstpersonwhoreportsit
tous. We alsoencouragereaderstocheckout theweb siteforUsing Algebraic
Geometry,whichisat
http://www.cs.amherst.edu/˜dac/uag.html
x PrefacetotheFirstEdition
Thissiteincludesupdatesanderratasheets,aswellaslinkstoothersitesofinter-
est.
Acknowledgments
We would like to thank everyone who sent us comments on initial drafts of the
manuscript.WeareespeciallygratefultothankSusanColley,AliciaDickenstein,
Ioannis Emiris, Tom Garrity, Pat Fitzpatrick, Gert-Martin Greuel, Paul Peder-
sen, Maurice Rojas, Jerry Shurman, Michael Singer, Michael Stanfield, Bernd
Sturmfels(andstudents),MossSweedler(andstudents),andWilandSchmalefor
especiallydetailedcommentsandcriticism.
We also gratefully acknowledge the support provided by National Science
FoundationgrantDUE-9666132, andthehelpandadviceaffordedbythemem-
bers of our AdvisoryBoard: Susan Colley, KeithDevlin, ArnieOstebee, Bernd
Sturmfels,andJimWhite.
November,1997 DavidCox
JohnLittle
DonalO’Shea
Contents
PrefacetotheSecondEdition v
PrefacetotheFirstEdition vii
1 Introduction 1
§1 PolynomialsandIdeals . . . . . . . . . . . . . . . . . . . . . . . 1
§2 MonomialOrdersandPolynomialDivision . . . . . . . . . . . . 6
§3 Gro¨bnerBases. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
§4 AffineVarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 SolvingPolynomialEquations 26
§1 SolvingPolynomialSystemsbyElimination . . . . . . . . . . . . 26
§2 Finite-DimensionalAlgebras . . . . . . . . . . . . . . . . . . . . 37
§3 Gro¨bnerBasisConversion . . . . . . . . . . . . . . . . . . . . . 49
§4 SolvingEquationsviaEigenvaluesandEigenvectors . . . . . . . 56
§5 RealRootLocationandIsolation . . . . . . . . . . . . . . . . . . 69
3 Resultants 77
§1 TheResultantofTwoPolynomials . . . . . . . . . . . . . . . . . 77
§2 MultipolynomialResultants. . . . . . . . . . . . . . . . . . . . . 84
§3 PropertiesofResultants . . . . . . . . . . . . . . . . . . . . . . . 95
§4 ComputingResultants. . . . . . . . . . . . . . . . . . . . . . . . 102
§5 SolvingEquationsviaResultants . . . . . . . . . . . . . . . . . . 114
§6 SolvingEquationsviaEigenvaluesandEigenvectors . . . . . . . 128
4 ComputationinLocalRings 137
§1 LocalRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
§2 MultiplicitiesandMilnorNumbers . . . . . . . . . . . . . . . . . 145
§3 TermOrdersandDivisioninLocalRings . . . . . . . . . . . . . 158
§4 StandardBasesinLocalRings . . . . . . . . . . . . . . . . . . . 174
§5 ApplicationsofStandardBases . . . . . . . . . . . . . . . . . . . 180
xi
xii Contents
5 Modules 189
§1 ModulesoverRings . . . . . . . . . . . . . . . . . . . . . . . . . 189
§2 MonomialOrdersandGro¨bnerBasesforModules. . . . . . . . . 207
§3 ComputingSyzygies . . . . . . . . . . . . . . . . . . . . . . . . 222
§4 ModulesoverLocalRings . . . . . . . . . . . . . . . . . . . . . 234
6 FreeResolutions 247
§1 PresentationsandResolutionsofModules . . . . . . . . . . . . . 247
§2 Hilbert’sSyzygyTheorem . . . . . . . . . . . . . . . . . . . . . 258
§3 GradedResolutions . . . . . . . . . . . . . . . . . . . . . . . . . 266
§4 HilbertPolynomialsandGeometricApplications . . . . . . . . . 280
7 Polytopes,Resultants,andEquations 305
§1 GeometryofPolytopes . . . . . . . . . . . . . . . . . . . . . . . 305
§2 SparseResultants . . . . . . . . . . . . . . . . . . . . . . . . . . 313
§3 ToricVarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
§4 MinkowskiSumsandMixedVolumes . . . . . . . . . . . . . . . 332
§5 Bernstein’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . 342
§6 ComputingResultantsandSolvingEquations . . . . . . . . . . . 357
8 PolyhedralRegionsandPolynomials 376
§1 IntegerProgramming . . . . . . . . . . . . . . . . . . . . . . . . 376
§2 IntegerProgrammingandCombinatorics. . . . . . . . . . . . . . 392
§3 MultivariatePolynomialSplines . . . . . . . . . . . . . . . . . . 405
§4 TheGro¨bnerFanofanIdeal . . . . . . . . . . . . . . . . . . . . 426
§5 TheGro¨bnerWalk . . . . . . . . . . . . . . . . . . . . . . . . . 436
9 AlgebraicCodingTheory 451
§1 FiniteFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
§2 Error-CorrectingCodes . . . . . . . . . . . . . . . . . . . . . . . 459
§3 CyclicCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
§4 Reed-SolomonDecodingAlgorithms. . . . . . . . . . . . . . . . 480
10 TheBerlekamp-Massey-SakataDecodingAlgorithm 494
§1 CodesfromOrderDomains . . . . . . . . . . . . . . . . . . . . . 494
§2 TheOverallStructureoftheBMSAlgorithm . . . . . . . . . . . 508
§3 TheDetailsoftheBMSAlgorithm . . . . . . . . . . . . . . . . . 522
References 533
Index 547
