Combinatorial Criteria for Gröbner Bases

259 Pages·2005·1.206 MB·English

by  John Edward Perry

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.
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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.
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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
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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
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