Table Of ContentLecture Notes in Mathematics 2121
Daniel Robertz
Formal
Algorithmic
Elimination
for PDEs
Lecture Notes in Mathematics 2121
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Daniel Robertz
Formal Algorithmic
Elimination for PDEs
123
Daniel Robertz
Plymouth University
School of Computing & Mathematics
Plymouth, UK
ISBN978-3-319-11444-6 ISBN978-3-319-11445-3(eBook)
DOI10.1007/978-3-319-11445-3
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2014952682
MathematicsSubjectClassification(2010):12H05,13N10, 16S32, 35C05
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Preface
Thismonographoriginatesinmyhabilitationtreatise.Thethemeistounderstand
thecorrespondencebetweensystemsofpartialdifferentialequationsandtheirana-
lyticsolutionsfromaformalviewpoint.Weconsiderontheonehandtheproblem
ofdeterminingthesetofanalyticsolutionstosuchasystemandontheotherhand
thatoffindingdifferentialequationswhosesetofsolutionscoincideswithagiven
set of analytic functions. Insight into one of these problems may advance the un-
derstanding of the other one. In particular, as shown in this monograph, symbolic
solvingofpartialdifferentialequationsprofitsfromthestudyoftheimplicitization
problemforcertainparametrizedsetsofanalyticfunctions.
Twoalgorithmsarefundamentalforthiswork.Janet’sandThomas’algorithms
transform a given system of linear or nonlinear partial differential equations into
an equivalent form allowing to determine all formal power series solutions in a
straightforwardway.Moreover,theseeffectivedifferentialalgebratechniquessolve
certaineliminationproblemswhenappliedinanappropriateway.Thefirstpartof
thismonographdiscussesthesemethodsandrelatedconstructionsindetail.Efficient
implementations of both algorithms in the computer algebra system Maple have
beendevelopedbytheauthorofthismonographandhiscolleagues.
Thesecondpartofthismonographaddressestheproblemoffindinganimplicit
descriptionintermsofdifferentialequationsofasetofanalyticfunctionswhichis
givenbyaparametrizationofacertainkind.Effectivemethodsofdifferentgener-
ality are developed to solve the differential elimination problems that arise in this
context. As a prominent application of these results it is demonstrated how some
familyofexactsolutionsoftheNavier-Stokesequationscanbecomputed.
I would like to thank several persons for many fruitful discussions about alge-
bra,aboutPDEs,andtheirconnection,andforsupportingmywork.Inparticular,I
wouldliketoexpressmygratitudetoProf.Dr.W.Plesken,Prof.Dr.J.Bemelmans,
Prof. Dr. C. Melcher, Prof. Dr. S. Walcher at RWTH Aachen and Priv.-Doz. Dr.
M.BarakatatTUKaiserslautern.ThanksgotoProf.Dr.V.P.Gerdt(JINRDubna)
and Dr. A. Quadrat (Inria Saclay), not only because I have been enjoying our col-
laborations,butalsoforreadingpartsofafirstversionofthismonograph.
v
vi Preface
InthecontextofworkonThomas’algorithm,IamgratefultomycolleaguesDr.
T.Ba¨chlerandDr.M.Lange-Hegermannfortheirhelp.Moreover,weprofitedvery
much from discussions with Prof. Dr. D. Wang (Paris 6), Prof. Dr. F. Boulier, and
Dr.F.Lemaire(bothatUniversite´ deLille)onthistopic.
Furthermore,IwouldliketothankallcolleaguesatLehrstuhlBfu¨rMathematik,
RWTH Aachen, for a very nice atmosphere, and, in particular, Mr. W. Krass for
linguisticadvice.
Finally, I am indebted to two anonymous referees for excellent suggestions on
anearlierversionofthismonograph.
Plymouth,England DanielRobertz
July2014
Contents
1 Introduction................................................... 1
2 FormalMethodsforPDESystems ............................... 5
2.1 Janet’sAlgorithm .......................................... 5
2.1.1 CombinatoricsofJanetDivision........................ 9
2.1.2 OreAlgebras........................................ 16
2.1.3 JanetBasesforOreAlgebras .......................... 21
2.1.4 ComparisonandComplexity........................... 37
2.1.5 TheGeneralizedHilbertSeries......................... 40
2.1.6 Implementations..................................... 54
2.2 ThomasDecompositionofDifferentialSystems ................. 57
2.2.1 SimpleAlgebraicSystems............................. 60
2.2.2 SimpleDifferentialSystems ........................... 88
2.2.3 TheGenericSimpleSystemforaPrimeIdeal ............104
2.2.4 ComparisonandComplexity...........................110
2.2.5 HilbertSeriesforSimpleDifferentialSystems............112
2.2.6 Implementations.....................................116
3 DifferentialEliminationforAnalyticFunctions....................119
3.1 Elimination ...............................................119
3.1.1 EliminationofVariablesusingJanetBases...............121
3.1.2 ComplexityReductionbymeansofaVeroneseEmbedding.128
3.1.3 EliminationofUnknownFunctionsusingJanetBases......134
3.1.4 EliminationviaThomasDecomposition .................139
3.1.5 CompatibilityConditionsforLinearSystems.............147
3.2 LinearDifferentialElimination ...............................160
3.2.1 ImplicitizationProblemforLinearParametrizations .......161
3.2.2 DifferentialEliminationusingLinearAlgebra ............165
3.2.3 ConstructingaRepresentation .........................178
3.2.4 AnnihilatorsforLinearParametrizations.................182
3.2.5 Applications ........................................186
vii
viii Contents
3.3 NonlinearDifferentialElimination ............................192
3.3.1 ImplicitizationProblemforMultilinearParametrizations ...194
3.3.2 EliminationMethodsforMultilinearParametrizations .....198
3.3.3 ImplicitizationusingKa¨hlerDifferentials ................207
3.3.4 ConstructingaRepresentation .........................216
3.3.5 Applications ........................................221
A BasicPrinciplesandSupplementaryMaterial .....................233
A.1 ModuleTheoryandHomologicalAlgebra......................233
A.1.1 FreeModules .......................................233
A.1.2 ProjectiveModulesandInjectiveModules ...............235
A.1.3 SyzygiesandResolutions .............................238
A.2 TheChainRuleofDifferentiation.............................243
A.3 DifferentialAlgebra ........................................247
A.3.1 BasicNotionsofDifferentialAlgebra ...................247
A.3.2 CharacteristicSets ...................................249
A.3.3 HomogeneousDifferentialPolynomials .................253
A.3.4 BasisTheoremandNullstellensatz......................256
References.........................................................259
ListofAlgorithms ..................................................273
ListofExamples ...................................................275
IndexofNotation...................................................277
Index .............................................................279
Chapter 1
Introduction
Differentialequationsareubiquitousinthemodelingofphenomenastudiedinthe
naturalsciences,intheirsimulation,andinengineeringapplications.Alotofmath-
ematicalresearchisdevotedtounderstandingmanydifferentaspectsofdifferential
equationsandtodevelopingmethodswhichdetermine(exactorapproximate)solu-
tionsofdifferentialequations.
Thismonographapproachespartialdifferentialequations(PDEs)fromtheview-
point of algebra and contributes algorithmic methods which allow to investigate
effectivelytherelationshipofsystemsofPDEsandtheirsetsofsolutions.Employ-
ing formal techniques, the focus is on polynomial differential equations and their
analyticsolutions.
Weborrowquiteafewconceptsfromalgebraicgeometry,whichisatheorythat
evolved over centuries from the continued quest to solve algebraic equations. The
contemporaryunderstandingofgeometricfactsinthiscontextisformulatedinterms
of polynomial ideals, the precise connection between geometry and algebra being
establishedbyHilbert’sNullstellensatz.
Whenever a set of points is given by a polynomial or rational parametrization,
aneliminationoftheparametersfromtheequationswhichexpressthecoordinates
ofthepointsyieldsequationsthataresatisfiedbythecoordinatesofeverypointof
theset.Ifthereexistsanimplicitdescriptionofthissetassolutionsetofasystem
ofpolynomial equations,then elimination constructssuch adescription.Since not
every such solution set admits a rational parametrization, the scope of the inverse
problemhastoberestrictedinsomesense,butmethodsareknownwhichconstruct
rationalparametrizationsforspecialvarieties.
Thismonographdevelopsalgorithmicmethodswhichaccomplishtheanalogous
elimination task for systems of polynomial partial differential equations and their
(complex)analyticsolutions.ItbuildsonworkbyCharlesRiquier,MauriceJanet,
JosephMillerThomas,JosephFelsRitt,EllisRobertKolchin,andothers,wholaid
thefoundationofdifferentialalgebra.
Agivenmultivariatepolynomial,whosecoefficientsareanalyticfunctions,isin-
terpretedasaparametrizationofasetofanalyticfunctions,i.e.,everyelementofthis
© Springer International Publishing Switzerland 2014 1
D. Robertz, Formal Algorithmic Elimination for PDEs,
Lecture Notes in Mathematics 2121, DOI 10.1007/978-3-319-11445-3_1
