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Combinatorial Criteria for Gröbner Bases PDF

259 Pages·2005·1.206 MB·English
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Abstract PERRY, JOHN EDWARD. Combinatorial Criteria for Gröbner Bases. (Under thedirectionofHoonHong.) Both the computation and the detection of Gröbner bases require a criterion that decides whether a set of polynomials is a Gröbner basis. The most funda- mental decision criterion is the reduction of all S-polynomials to zero. However, S-polynomial reduction is expensive in terms of time and storage, so a number ofresearchers haveinvestigatedthe questionof whenwecan avoidS-polynomial reduction. Certainresultscanbeconsidered“combinatorial”,becausetheyarecri- teria on the leading terms, which are determined by integers. Our research builds ontheseresults;thisthesispresentscombinatorialcriteriaforGröbnerbases. ThefirstpartofthisthesisreviewstherelevantliteratureonGröbnerbasesand skippingS-polynomialreduction. Thesecondpartconsiderscriteriaforskippinga fixednumberofS-polynomialreductions. Thefirsttwotheoremsofparttwoshow howtoapplyBuchberger’scriteriatoobtainnecessaryandsufficientconditionsfor skipping all S-polynomial reductions, and for skipping all but one S-polynomial reductions. The third theorem considers the question of skipping all but two S- polynomial reductions; we have found that this problem requires new criteria on leading terms. We provide one new criterion that solves this problem for a set of threepolynomials;forlargersets,theproblemremainsopen. The final part of this thesis considers Gröbner basis detection. After a brief review of a previous result that requires S-polynomial reduction, we provide a new result which takes a completely different approach, avoiding S-polynomial reductioncompletely. Throughout the latter two parts, we provide some statistical analysis and ex- perimentalresults. 8thApril2005 COMBINATORIAL CRITERIA FOR GRÖBNER BASES BY JOHN EDWARD PERRY, III A DISSERTATION SUBMITTED TO THE GRADUATE FACULTY OF NORTH CAROLINA STATE UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS RALEIGH, NORTH CAROLINA APRIL 2005 APPROVED BY: H. HONG E. KALTOFEN CHAIR OF ADVISORY COMMITTEE A. SZANTO M. SINGER Dedication ANonnoFelice: guardandoilnipotecheleggeva,videunprofessore.1 1Italian: ToNonnoFelice: helookedathisgrandsonwhowasreading,andsawaprofessor. ii Biography JohnPerry(III)wasborninlate1971,inthecityofNewportNews,Virginia. His parents are John Perry (Jr.), originally of Hampton, Virginia, and Maria Leboffe, originally of Gaeta, Italy. Following graduation from Warwick high school, John enteredMarymountUniversity. Hegraduatedin1993withhisB.S.inmathematics and mathematics education. John subsequently earned an M.S. in mathematics fromNorthernArizonaUniversity. Itwasatthistimethathebegantoexperiment with mathematics on the computer; one program that implemented the Runge- Kutta interpolation to graph differential equations is available on-line for Amiga computers: see[Per94]. After earning his master’s degree, John returned to Virginia and taught math- ematics at Franklin County High School. Two years later, he volunteered for the Catholic priesthood, but withdrew from seminary in December, 1998. He applied, andwasaccepted,toNorthCarolinaStateUniversity. Heenteredinthefallof1999 andspentthenextsixyearsworkingonhisdoctorate. iii Acknowledgments First,ImustexpressmyprofoundgratitudetothecitizensofthestateofNorth Carolina, whose system of public universities is among the finest in the world. I likewisethankthecitizensoftheUnitedStates,whosupportedmyresearchinpart withNSFgrant53344. I could never have hoped to be a doctoral student of mathematics at North Carolina State University without both the inspiration and the instruction of past teachers and professors. There is insufficient space to name them all, but I would be remiss if I failed to name Neil Drummond, Vanessa Job, Judy Green, Adrian Riskin(orAdrienRiskin,dependingonhismood),JonathanHargis,andLawrence Perko. IalsothankthemathematicsdepartmentatNorthCarolinaStateUniversity,es- peciallythecomputeralgebragroup,fromwhomIlearnedanimmenseamount. In particular, my committee(Hoon Hong, ErichKaltofen, Michael Singer, andAgnes Szanto)taughtexcellentclassesandlookedatseveralincompletedraftsofthistext. iv My advisor, Hoon Hong, spent countless hours guiding my research, teaching mehowtoaskquestions,howtoanswerthem,andhowtorealizewhentheques- tion might be too hard. His advice and encouragement were invaluable. His wife provided a number of delicious meals on many of the occasions that I passed the hourswithherhusbandhunchedoverasheetofpaper;ononeoccasionshedrove metoacarmechanic. Erich Kaltofen provided a number of thoughtful conversations, not all about mathematics. One of these conversations (on how mathematicians create ideas) helped me realize that it was not yet time to abandon hope. He was always ready withadviceandinformation. BrunoBuchbergersuggestedthestatisticalinvestigationsoftheresultsthatap- pear at the end of chapter 6. Our discussions on the results of that chapter went a longwaytowardsdeepeningmyunderstandingofthem. My officemates Chris Kuster, Alexey Ovchinnikov, Scott Pope, and George Yuhasz were especially generous with their time and made a number of helpful suggestionsandcontributionsonpapers,presentations,andthisthesis. Richard Schugart, my roommate of these past five years, has had to suffer my quirky, a-social personality. He has put up with me much better than I proba- bly would. His mother was very generous with cookies, for which I’m not sure whetherIowehergratitudeorgrumbling–alas,mywaistlinehasexpanded. v My parents and my brothers have patiently endured my dissatisfaction with personal and professional imperfections for thirty-three years now, and they have never failed to challenge me to do my best. I come from a family privileged not bywealth,butbythecontentofitscharacter,tosaynothingofthecharactersinits contents. Ardisciespera!2 Gal(cid:31) nagradila ulybkoi$ na sem~ roz. 3 Finally: laustibi,lucislargitorsplendide.4 2AsayingofmyItaliangrandfatherandhisbrother: Dareandhope! 3Galyarewardedsevenroseswithasmile. 4AdaptedfromamedievalLatinhymn: Praisetoyou,Ogleaminggiveroflight. vi Table of Contents ListofFigures ix ListofTables xi ListofSymbols xiii Part1. BackgroundMaterial 1 Chapter1. AnintroductiontoGröbnerbases 2 1.1. Gröbnerbases: analogy 2 1.2. Gröbnerbases: definition 7 1.3. Gröbnerbases: decision 22 1.4. SomepropertiesofrepresentationsofS-polynomials 48 Chapter2. SkippingS-polynomialreductions 54 2.1. ABottleneck 54 2.2. SkippingS-polynomialReductions 55 2.3. CombinatorialCriteriaonLeadingTerms 58 2.4. TheBuchbergerCriteria 59 2.5. TermDiagrams 71 Part2. NewcombinatorialcriteriaforskippingS-polynomialreduction 77 Chapter3. Outlineofparttwo 78 3.1. Buchberger’scriteriarevisited 78 3.2. Formalstatementoftheproblemconsideredinparttwo 80 3.3. Outlineoftheremainderofparttwo 82 3.4. Aninvaluablelemma 83 Chapter4. SkippingallS-polynomialreductions 86 4.1. Problem 86 4.2. Result 87 4.3. Applicationofresult 89 Chapter5. SkippingallbutoneS-polynomialreductions 101 vii 5.1. Problem 101 5.2. Result 102 5.3. Applicationofresult 108 Chapter 6. Skipping all but two S-polynomial reductions (case for three polynomials) 120 6.1. Problem 120 6.2. Result 123 6.3. Applicationofresult 148 Part3. ANewCriterionforGröbnerBasisDetectionofTwoPolynomials 167 Chapter7. Gröbnerbasisdetection: introduction 168 7.1. Introductiontotheproblem 168 7.2. Matrixrepresentationsoftermorderings 171 7.3. Solvingforadmissiblematrices 186 Chapter8. Gröbnerbasisdetection: generalsolutionbypolytopes 200 8.1. Problem 200 8.2. Polytopes,PolyhedraandMinkowskiSums 201 8.3. Result 205 Chapter9. Gröbnerbasisdetectionoftwopolynomialsbyfactoring 209 9.1. Problem 209 9.2. Result 209 9.3. Applicationofresult 213 Conclusion 235 Bibliography 237 Index 241 viii

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