The Gröbner Cover

285 Pages·2018·6.136 MB·English

by  Antonio Montes

Algorithms and Computation in Mathematics 27
Antonio Montes
The
Gröbner Cover
Algorithms and Computation
•
in Mathematics Volume 27
Editors
D.Eisenbud M.F.Singer
B.Sturmfels M.Braverman
B.Viray
Moreinformationaboutthisseriesathttp://www.springer.com/series/3339
Antonio Montes
¨
The Grobner Cover
123
AntonioMontes
UniversitatPolite`cnicadeCatalunya
Barcelona,Spain
ISSN1431-1550
AlgorithmsandComputationinMathematics
ISBN978-3-030-03903-5 ISBN978-3-030-03904-2 (eBook)
https://doi.org/10.1007/978-3-030-03904-2
LibraryofCongressControlNumber:2018962627
MathematicsSubjectClassification(2010):13-02,13-04,11C08,13B25,13A15,13F20,13P10,13P15,
13P25,68T15,03B35
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ToRosa
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Preface
The role of computersin mathematicshas createda new perspectivein which the
development of algorithms becomes central. This was the reason for developing
Computer Algebra Software systems (CAS). In 1965, Bruno Buchberger in his
thesis [14], introduced Gröbner Basis (GB) and his famous algorithm providing
the canonicalrepresentationof ordinarypolynomialsystems. It allowsforsolving
problemsconcerningpolynomialequations.Manyproblemsinmathematics,engi-
neeringandotherscientificdomainscanbemodelledbypolynomialequations.
Since then, Gröbner Basis has become a classical subject of the Computer
AlgebracurriculumatanundergraduatelevelduetotheexcellenttextbooksofCox,
Little,O’Shea [25],Becker-Weispfenning [6],Adams-Loustaunau [2],Eisenbud
[30],andBuchberger’salgorithmhasbeenimplementedinmanyCASsystems.
Between the CAS systems implementingGröbner bases, Singular [62], devel-
opedbyGreuel,PfisterandSchönemann,atKaiserslauternTechnicalUniversityis
oneofthemostefficient.Thebooks [28,39],contributealsotoitsrelevance.
Ordinary Gröbner Bases can solve polynomial systems that depend only on
variables but not on parameters. As the polynomial equations appearing in many
technical problems contain parameters, a new approach was necessary. Volker
Weispfenning [83] introducedComprehensiveGröbnerSystems(CGS) andBasis
(GCB) to tackle polynomial systems with parameters. Since then, many other
mathematicianshavedevelopedandimprovednewalgorithmsforthesamepurpose
[41,42,42,54–56,66,69,75,84].
In 2007 Michael Wibmer, in his thesis [85], proved a central theorem for
parametricpolynomialsystems.Usingit,in2010,WibmerandI [63]developedthe
GröbnerCover,whichisthecanonicalalgorithmforsolvingparametricpolynomial
systemsandistheanalogueoftheReducedGröbnerbasisforparametricsystems.
Starting in 2008, the author with the priceless help of Hans Schönemann
implemented the SINGULAR “grobcov.lib” library. Since then, the library has
continuously been improved to solve new kinds of problems in which parametric
polynomialsystemsarecentral.
ThisbookgivesacompletedescriptionoftheCanonicalGröbnerCover.Italso
serves as a User’s Manual for the SINGULAR “grobcov.lib” library, as it contains
vii
viii Preface
manyexamplesandcomments.Allthealgorithmsdescribedareimplementedinthe
library.
It is assumed that the reader is familiar with ordinary Gröbner basis theory of
multivariatepolynomials.
Afirstpreliminarychapterofthebookisdevotedtosomebasicresults(without
proof or completeness) of ordinary Gröbner basis theory, which are widely used
in this text. The objective is to introduce the notations and to recall some central
elements of the theory. However, it should not, of course, be considered as a
complete description of ordinary Gröbner basis theory. We also present a simple
exampleoftheuseoftheGröbnerCover(GC)togiveanideaofitsusefulness.
Next, the book is divided into two parts. Part I: Theory has four chapters:
Chaps.2, 3, 4, contain the basic tools needed in the construction of the Gröbner
Cover,whichisdescribedinChap.5.
Chapter2isdevotedtoconstructiblesets.IncollaborationwithJosepM.Brunat,
we have recently established the canonical form of constructible sets [13]. It is
described in this chapter and used later in the applications to loci computations.
Nevertheless, for the Gröbner Cover it is not necessary to use the whole canon-
ical algorithm, because Wibmer’s Theorem [85] ensures that, for homogeneous
polynomials, the union of all the segments of a Comprehensive Gröbner System
corresponding to the same set of lpp (leading power products) is locally closed.
Thus,itsufficestouseasimpleralgorithmcalledLCUNION.
Chapter 3 describes the algorithm for computing a disjoint and reduced Com-
prehensive Gröbner System (CGS), which is the previous step for the GC. When
the GC was developed in 2008, we used our own algorithm, called BUILDTREE,
which is described in the Appendix. In 2010, Kapur, Sun, Wang [43] designed
a better algorithm for this purpose, which is described in this chapter; this is the
CGSalgorithmincludedinthecurrentSINGULARlibraryaspartoftheinstruments
neededfortheGC.
Chapter4describesI-regularfunctions,whicharefunctionsdefinedonalocally
closed set, and determined by one or more polynomials. They are essential for
Wibmer’sTheorem [85].
Chapter5isthecentralpartdescribingtheGröbnerCover [63].Init,Wibmer’s
Theorem and the existence of the Canonical Gröbner Cover are proved, and the
algorithmsprovided.ThetheoryandalgorithmsdevelopedinChaps.2,3and4are
usedforthispurpose.Inafirstreadingonecangodirectlytothischapterandconsult
thepreviouschapterswhenneeded.
EventhoughtherearemanypossibledifferentapplicationsoftheGC, we have
chosen only three for which the Gröbner Cover gives new insight, that we have
developedincollaborationwithM.A.Abánades,F.Botana,T.Recio.
ThesethreeapplicationsarethecontentsofPartII:Applications.
Chapter 6 is an application to Automatic Deduction of Geometric Theorems
(ADGT).ItwasoriginallydevelopedincollaborationwithT.Recio [60,61],andit
hasbeenimprovedinthischapter.
Chapter7isdevotedtotheGCinDynamicalGeometry(DG)forthecomputation
andclassificationofgeometricalloci.Thisisaninterestingapplicationthathasbeen
Preface ix
testedinthewidelyknownDGsoftwareGeoGebra.M.A.Abánades,F.Botanaand
Z.KovacshavedevelopedapplicationstoGeoGebra,andtogetherwithT.Recioand
theauthor [1],wedefinedataxonomyforgeometricallociusingGC.Inthischapter,
we also extendthe definitionsand theoremsfor locus taxonomyto n-dimensional
space.
Chapter 8 is devotedto envelopesof familiesof hypersurfacesthatcan also be
used in DG. A generalized definition of envelope is given that leads to solving
newkindsofproblems.Newalgorithmsforobtainingtheassociatedtangentfamily
element at a certain point of the envelope, as well as determining all the family
elementsthatpassatacertainpointoftheenvelope,aredeveloped.Theycanhelp
analyze the structure of the real algebraic envelope components. This is a new
developmentthathasbeenimprovedandrewritteninthisbook.F.Botana,Z.Kovacs
andT.Recio,havealsodevelopedapplicationstoGeoGebra.
Algorithmsandcommandsofthelibraryareincludedinthebookwithdifferent
fonts. For example, GROBCOV is the algorithm and grobcov the implemented
commandofthe library.Noteveryalgorithmhasa correspondingcommandsince
some algorithms are auxiliary and not public. Moreover, their names can be
slightly different. For example EXTEND and extendpoly and extendGC. All
thealgorithmsaredescribedbutonlytheimplementedcommandsofthelibraryare
usedintheexamples.
Barcelona,Spain AntonioMontes
Acknowledgements
Iamindebtedandgratefultomanypeople.First,toProf.TomásRecio(University
of Cantabria, Spain), who initiatiated of the Spanish group of Computer Algebra
to which I was invited from the beginningin 1994 during the MEGA’94 meeting
(Effective Methods in Algebraic Geometry) at Cantabria University. Since then, I
havereceivedcontinuousscientificsupportfromhim.Wehavealsocollaboratedin
thedevelopmentofinterestingapplicationsoftheGröbnerCoverthataredescribed
in this book. We founded, together with other Spanish colleagues, the EACA
committee(EncuentrosdeAlgebraComputacionalyAplicaciones)andtheEACA
network, which has become a met in the international community of Computer
Algebra.InMEGA’94I also metProf.VolkerWeispfenningofPassau University,
who is the nextperson to whomI am indebted.I was applyingGröbnerbases for
electricalnetworks,andhesuggestedusinghisrecentlyintroducednew algorithm
forparametricpolynomialsystems,whichiswheremyinterestinthesubjectcomes
from.HeinvitedmetoPassauUniversityandweestablishedawarmrelationship.
In the next years, I developednew algorithms for improvingthe Weispfenning
CGS algorithm. My Ph.D. student Montserrat Manubens collaborated in this
development. Together we developed first our BuildTree algorithm for obtaining
aCGS,andthenwedesignedanotheralgorithmwhich,takingitasastartingpoint,
combinessegmentswiththesamesetoflppallowingforacommonbasis.Thusthe
completealgorithmgeneratesa verycompactdiscussion [56]. I am also indebted
toProf.YosukeSatoofTokyoUniversitywhoinvitedmetogotoJapanandKorea
todiscussthesubject.
Most especially, I am indebted to Michael Wibmer, who proposed his central
Theorem in 2007. He suggested that I develop an algorithm to implement it. The
result was the Gröbner Cover (GC) [63], which starting in 2008, we developed
at Heidelberg University. Without his ideas, it would not have been possible to
developtheGC.In2008,IwasinvitedbyProf.GertMartinGreueltoKaiserslautern
TechnicalUniversityfora sabbaticalperiod,whereI collaboratedwithProf.Hans
Schönemann in the implementation of the GC algorithm using SINGULAR and
creating the “grobcov.lib” library. The library has evolved, and has now become
arobustpackage [62].In2016IwasoncemoreinvitedtoKaiserslauternbyProf.
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