Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra PDF
Preview Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra
Undergraduate Texts in Mathematics Editors S.Axler F.W.Gehring K.A.Ribet Undergraduate Texts in Mathematics Abbott:Understanding Analysis. Devlin:The Joy of Sets:Fundamentals Anglin:Mathematics:A Concise History of Contemporary Set Theory.Second and Philosophy. edition. Readings in Mathematics. Dixmier:General Topology. Anglin/Lambek:The Heritage ofThales. Driver:Why Math? Readings in Mathematics. Ebbinghaus/Flum/Thomas:Mathematical Apostol:Introduction to Analytic Number Logic.Second edition. Theory.Second edition. Edgar:Measure,Topology,and Fractal Armstrong:Basic Topology. Geometry. Armstrong:Groups and Symmetry. Elaydi:An Introduction to Difference Axler:Linear Algebra Done Right.Second Equations.Third edition. edition. Erdõs/Surányi:Topics in the Theory of Beardon:Limits:A New Approach to Real Numbers. Analysis. Estep:Practical Analysis on One Variable. Bak/Newman:Complex Analysis.Second Exner:An Accompaniment to Higher edition. Mathematics. Banchoff/Wermer:Linear Algebra Through Exner:Inside Calculus. Geometry.Second edition. Fine/Rosenberger:The Fundamental Theory Berberian:A First Course in Real Analysis. ofAlgebra. Bix:Conics and Cubics:A Concrete Fischer:Intermediate Real Analysis. Introduction to Algebraic Curves. Flanigan/Kazdan:Calculus Two:Linear and Brèmaud:An Introduction to Probabilistic Nonlinear Functions.Second edition. Modeling. Fleming:Functions ofSeveral Variables. Bressoud:Factorization and Primality Testing. Second edition. Bressoud:Second Year Calculus. Foulds:Combinatorial Optimization for Readings in Mathematics. Undergraduates. Brickman:Mathematical Introduction to Foulds:Optimization Techniques:An Linear Programming and Game Theory. Introduction. Browder:Mathematical Analysis:An Franklin:Methods ofMathematical Introduction. Economics. Buchmann:Introduction to Cryptography. Frazier:An Introduction to Wavelets Second edition. Through Linear Algebra. Buskes/van Rooij:Topological Spaces:From Gamelin:Complex Analysis. Distance to Neighborhood. Ghorpade/Limaye:A Course in Calculus and Callahan:The Geometry ofSpacetime:An Real Analysis. Introduction to Special and General Gordon:Discrete Probability. Relavitity. Hairer/Wanner:Analysis by Its History. Carter/van Brunt:The Lebesgue– Stieltjes Readings in Mathematics. Integral:A Practical Introduction. Halmos:Finite-Dimensional Vector Spaces. Cederberg:A Course in Modern Second edition. Geometries.Second edition. Halmos:Naive Set Theory. Chambert-Loir:A Field Guide to Algebra Hämmerlin/Hoffmann:Numerical Childs:A Concrete Introduction to Higher Mathematics. Algebra.Second edition. Readings in Mathematics. Chung/AitSahlia:Elementary Probability Harris/Hirst/Mossinghoff:Combinatorics Theory:With Stochastic Processes and an and Graph Theory. Introduction to Mathematical Finance. Hartshorne:Geometry:Euclid and Fourth edition. Beyond. Cox/Little/O’Shea:Ideals,Varieties,and Hijab:Introduction to Calculus and Algorithms.Third edition.(2007) Classical Analysis. Croom:Basic Concepts ofAlgebraic Hilton/Holton/Pedersen:Mathematical Topology. Reflections:In a Room with Many Cull/Flahive/Robson:Difference Equations. Mirrors. From Rabbits to Chaos. Hilton/Holton/Pedersen:Mathematical Curtis:Linear Algebra:An Introductory Vistas:From a Room with Many Approach.Fourth edition. Windows. Daepp/Gorkin:Reading,Writing,and Proving:A Closer Look at Mathematics. (continued after index) David Cox John Little Donal O’Shea Ideals, Varieties, and Algorithms An Introduction to Computational Algebraic Geometry and Commutative Algebra Third Edition David Cox John Little Donal O’Shea Department ofMathematics Department ofMathematics Department ofMathematics and Computer Science College ofthe Holy Cross and Statistics Amherst College Worcester,MA 01610-2395 Mount Holyoke College Amherst,MA 01002-5000 USA South Hadley,MA 01075-1493 USA USA Editorial Board S.Axler F.W.Gehring K.A.Ribet Mathematics Department Mathematics Department Department ofMathematics San Francisco State East Hall University ofCalifornia University University ofMichigan at Berkeley San Francisco,CA 94132 Ann Arbor,MI 48109 Berkeley,CA 94720-3840 USA USA USA Mathematics Subject Classification (2000):14-01,13-01,13Pxx Library ofCongress Control Number:2006930875 ISBN-10:0-387-35650-9 e-ISBN-10:0-387-35651-7 ISBN-13:978-0-387-35650-1 e-ISBN-13:978-0-387-35651-8 Printed on acid-free paper. © 2007,1997,1992 Springer Science+Business Media,LLC All rights reserved.This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media,LLC,233 Spring Street, New York,NY 10013,USA),except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation,computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication oftrade names,trademarks,service marks,and similar terms,even if they are not identified as such,is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com P1:OTE/SPH P2:OTE/SPH QC:OTE/SPH T1:OTE SVNY310-COX January5,2007 18:17 ToElaine, forherloveandsupport. D.A.C. Tomymotherandthememoryofmyfather. J.B.L. ToMaryandmychildren. D.O’S. v P1:OTE/SPH P2:OTE/SPH QC:OTE/SPH T1:OTE SVNY310-COX January5,2007 18:17 Preface to the First Edition Wewrotethisbooktointroduceundergraduatestosomeinterestingideasinalgebraic geometryandcommutativealgebra.Untilrecently,thesetopicsinvolvedalotofabstract mathematics and were only taught in graduate school. But in the 1960s, Buchberger andHironakadiscoverednewalgorithmsformanipulatingsystemsofpolynomialequa- tions. Fueled by the development of computers fast enough to run these algorithms, thelasttwodecadeshaveseenaminorrevolutionincommutativealgebra.Theability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand, and has changed the practice of much research in algebraic geometry. This has also enhanced the impor- tanceofthesubjectforcomputerscientistsandengineers,whohavebeguntousethese techniquesinawholerangeofproblems. Itisourbeliefthatthegrowingimportanceofthesecomputationaltechniqueswar- rantstheirintroductionintotheundergraduate(andgraduate)mathematicscurriculum. Manyundergraduatesenjoytheconcrete,almostnineteenth-century,flavorthatacom- putationalemphasisbringstothesubject.Atthesametime,onecandosomesubstan- tial mathematics, including the Hilbert Basis Theorem, Elimination Theory, and the Nullstellensatz. Themathematicalprerequisitesofthebookaremodest:thestudentsshouldhavehad acourseinlinearalgebraandacoursewheretheylearnedhowtodoproofs.Examples ofthelattersortofcourseincludediscretemathandabstractalgebra.Itisimportantto notethatabstractalgebraisnotaprerequisite.Ontheotherhand,ifallofthestudents havehadabstractalgebra,thencertainpartsofthecoursewillgomuchmorequickly. Thebookassumesthatthestudentswillhaveaccesstoacomputeralgebrasystem. AppendixCdescribesthefeaturesofAXIOM,Maple,Mathematica,andREDUCEthat aremostrelevanttothetext.Wedonotassumeanypriorexperiencewithacomputer. However,manyofthealgorithmsinthebookaredescribedinpseudocode,whichmay beunfamiliartostudentswithnobackgroundinprogramming.AppendixBcontainsa carefuldescriptionofthepseudocodethatweuseinthetext. Inwritingthebook,wetriedtostructurethematerialsothatthebookcouldbeused inavarietyofcourses,andatavarietyofdifferentlevels.Forinstance,thebookcould serveasabasisofasecondcourseinundergraduateabstractalgebra,butwethinkthat it just as easily could provide a credible alternative to the first course. Although the vii P1:OTE/SPH P2:OTE/SPH QC:OTE/SPH T1:OTE SVNY310-COX January5,2007 18:17 viii PrefacetotheFirstEdition book is aimed primarily at undergraduates, it could also be used in various graduate courses,withsomesupplements.Inparticular,beginninggraduatecoursesinalgebraic geometryorcomputationalalgebramayfindthetextuseful.Wehope,ofcourse,that mathematiciansandcolleaguesinotherdisciplineswillenjoyreadingthebookasmuch asweenjoyedwritingit. Thefirstfourchaptersformthecoreofthebook.Itshouldbepossibletocoverthem ina14-weeksemester,andtheremaybesometimeleftoverattheendtoexploreother partsofthetext.Thefollowingchartexplainsthelogicaldependenceofthechapters: 1 2 3 4 6 8 5 7 9 Seethetableofcontentsforadescriptionofwhatiscoveredineachchapter.Asthe chart indicates, there are a variety of ways to proceed after covering the first four chapters.Also,atwo-semestercoursecouldbedesignedthatcoverstheentirebook. Forinstructorsinterestedinhavingtheirstudentsdoanindependentproject,wehave includedalistofpossibletopicsinAppendixD. It is a pleasure to thank the New England Consortium for Undergraduate Science Education (and its parent organization, the Pew Charitable Trusts) for providing the major funding for this work. The project would have been impossible without their support.VariousaspectsofourworkwerealsoaidedbygrantsfromIBMandtheSloan Foundation,theAlexandervonHumboldtFoundation,theDepartmentofEducation’s FIPSEprogram,theHowardHughesFoundation,andtheNationalScienceFoundation. Wearegratefulfortheirhelp. WealsowishtothankcolleaguesandstudentsatAmherstCollege,GeorgeMason University,HolyCrossCollege,MassachusettsInstituteofTechnology,MountHolyoke College,SmithCollege,andtheUniversityofMassachusettswhoparticipatedincour- sesbasedonearlyversionsofthemanuscript.Theirfeedbackimprovedthebookconsi- derably.Manyothercolleagueshavecontributedsuggestions,andwethankyouall. Corrections,commentsandsuggestionsforimprovementarewelcome! November1991 DavidCox JohnLittle DonalO’Shea P1:OTE/SPH P2:OTE/SPH QC:OTE/SPH T1:OTE SVNY310-COX January5,2007 18:17 Preface to the Second Edition In preparing a new edition of Ideals, Varieties, and Algorithms, our goal was to correct some of the omissions of the first edition while maintaining the readabil- ity and accessibility of the original. The major changes in the second edition are as follows: (cid:2) Chapter2:AbetteracknowledgementofBuchberger’scontributionsandanimproved proofoftheBuchbergerCriterionin§6. (cid:2) Chapter5:Animprovedboundonthenumberofsolutionsin§3andanew§6which completestheproofoftheClosureTheorembeguninChapter3. (cid:2) Chapter8:AcompleteproofoftheProjectionExtensionTheoremin§5andanew §7whichcontainsaproofofBezout’sTheorem. (cid:2) AppendixC:anewsectiononAXIOMandanupdateonwhatwesayaboutMaple, Mathematica,andREDUCE. Finally,wefixedsometypographicalerrors,improvedandclarifiednotation,andup- datedthebibliographybyaddingmanynewreferences. Wealsowanttotakethisopportunitytoacknowledgeourdebttothemanypeople whoinfluencedusandhelpedusinthecourseofthisproject.Inparticular,wewould liketothank: (cid:2) DavidBayerandMoniqueLejeune-Jalabert,whosethesisBAYER(1982)andnotes (cid:2) LEJEUNE-JALABERT(1985)firstacquainteduswiththiswonderfulsubject. Frances Kirwan, whose book KIRWAN (1992) convinced us to include Bezout’s TheoreminChapter8. (cid:2) StevenKleiman,whoshowedushowtoprovetheClosureTheoreminfullgenerality. HisproofappearsinChapter5. (cid:2) MichaelSinger,whosuggestedimprovementsinChapter5,includingthenewPropo- sition8of§3. (cid:2) Bernd Sturmfels, whose book STURMFELS (1993) was the inspiration for Chapter7. Therearealsomanyindividualswhofoundnumeroustypographicalerrorsandgave usfeedbackonvariousaspectsofthebook.Wearegratefultoyouall! ix P1:OTE/SPH P2:OTE/SPH QC:OTE/SPH T1:OTE SVNY310-COX January5,2007 18:17 x PrefacetotheSecondEdition Aswiththefirstedition,wewelcomecommentsandsuggestions,andwepay$1for everynewtypographicalerror.Foralistoferrorsandotherinformationrelevanttothe book,seeourwebsitehttp://www.cs.amherst.edu/∼dac/iva.html. October1996 DavidCox JohnLittle DonalO’Shea
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