Asymptotic Differential Algebra and Model Theory of Transseries

874 Pages·2017·5.887 MB·English

by  Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven

Annals of Mathematics Studies
Number 195

Asymptotic Differential Algebra
and Model Theory of Transseries
Matthias Aschenbrenner
Lou van den Dries
Joris van der Hoeven
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2017
Copyright©2017byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
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LibraryofCongressCataloging-in-PublicationData
Names: Aschenbrenner, Matthias, 1972– | van den Dries, Lou | Hoeven, J. van der
(Joris)
Title: Asymptoticdifferentialalgebraandmodeltheoryoftransseries/
MatthiasAschenbrenner,LouvandenDries,JorisvanderHoeven.
Description: Princeton: PrincetonUniversityPress,2017. |Series: Annalsofmathe-
maticsstudies;number195|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2017005899|ISBN9780691175423(hardcover: alk.paper)|
ISBN9780691175430(pbk. : alk.paper)
Subjects: LCSH: Series, Arithmetic. | Divergent series. | Asymptotic expansions. |
Differentialalgebra.
Classification: LCC QA295 .A87 2017 | DDC 512/.56–dc23 LC record available at
https://lccn.loc.gov/2017005899
BritishLibraryCataloging-in-PublicationDataisavailable
Thepublisherwouldliketoacknowledgetheauthorsofthisvolumeforprovidingthe
camera-readycopyfromwhichthisbookwasprinted.
ThisbookhasbeencomposedinLATEX.
Printedonacid-freepaper. ∞
10987654321
Had the apparatus [of transseries and analyzable functions] been
introducedforthesolepurposeofsolvingDulac’s“conjecture,”one
might legitimately question the wisdom and cost-effectiveness of
such massive investment in new machinery. However, [these no-
tions] have many more applications, actual or potential, especially
inthestudyofanalyticsingularities. Buttheirchiefattractionisper-
hapsthatofgivingconcrete,ifpartial,shapetoG.H.Hardy’sdream
of an all-inclusive, maximally stable algebra of “totally formalizable
functions.”
—JeanÉcalle,SixLecturesonTransseries,AnalysableFunctionsandthe
ConstructiveProofofDulac’sConjecture.
The virtue of model theory is its ability to organize succinctly the
sortoftiresomealgebraicdetailsassociatedwitheliminationtheory.
—GeraldSacks,TheDifferentialClosureofaDifferentialField.
Les analystes p-adiques se fichent tout autant que les géomètres
algébristes ..., des gammes à plus soif sur les valuations com-
posées,lesgroupesordonnésbaroques,sous-groupespleinsdes-
ditsetquesais-je. Cesgammesméritenttoutauplusd’enrichirles
exercicesdeBourbaki,tantquepersonnenes’ensert.
—AlexanderGrothendieck,lettertoSerredatedOctober31,1961.
I don’t like either writing or reading two-hundred page papers. It’s
notmyideaoffun.
—JohnH.Conway, quotedinGeniusatPlay: TheCuriousMindofJohn
HortonConway bySiobhanRoberts.

Contents
Preface xiii
ConventionsandNotations xv
Leitfaden xvii
DramatisPersonæ xix
IntroductionandOverview 1
ADifferentialFieldwithNoEscape 1
StrategyandMainResults 10
Organization 21
TheNextVolume 24
FutureChallenges 25
AHistoricalNoteonTransseries 26
1 SomeCommutativeAlgebra 29
1.1 TheZariskiTopologyandNoetherianity 29
1.2 RingsandModulesofFiniteLength 36
1.3 IntegralExtensionsandIntegrallyClosedDomains 39
1.4 LocalRings 43
1.5 Krull’sPrincipalIdealTheorem 50
1.6 RegularLocalRings 52
1.7 ModulesandDerivations 55
1.8 Differentials 59
1.9 DerivationsonFieldExtensions 67
2 ValuedAbelianGroups 70
2.1 OrderedSets 70
2.2 ValuedAbelianGroups 73
2.3 ValuedVectorSpaces 89
2.4 OrderedAbelianGroups 98
viii CONTENTS
3 ValuedFields 110
3.1 ValuationsonFields 110
3.2 PseudoconvergenceinValuedFields 126
3.3 HenselianValuedFields 136
3.4 DecomposingValuations 157
3.5 ValuedOrderedFields 171
3.6 SomeModelTheoryofValuedFields 179
3.7 TheNewtonTreeofaPolynomialoveraValuedField 186
4 DifferentialPolynomials 199
4.1 DifferentialFieldsandDifferentialPolynomials 199
4.2 DecompositionsofDifferentialPolynomials 209
4.3 OperationsonDifferentialPolynomials 214
4.4 ValuedDifferentialFieldsandContinuity 221
4.5 TheGaussianValuation 227
4.6 DifferentialRings 231
4.7 DifferentiallyClosedFields 237
5 LinearDifferentialPolynomials 241
5.1 LinearDifferentialOperators 241
5.2 Second-OrderLinearDifferentialOperators 258
5.3 DiagonalizationofMatrices 264
5.4 SystemsofLinearDifferentialEquations 270
5.5 DifferentialModules 276
5.6 LinearDifferentialOperatorsinthePresenceofaValuation 285
5.7 CompositionalConjugation 290
5.8 TheRiccatiTransform 298
5.9 Johnson’sTheorem 303
6 ValuedDifferentialFields 310
6.1 AsymptoticBehaviorofv 311
P
6.2 AlgebraicExtensions 314
6.3 ResidueExtensions 316
6.4 TheValuationInducedontheValueGroup 320
6.5 AsymptoticCouples 322
6.6 DominantPart 325
6.7 TheEqualizerTheorem 329
6.8 EvaluationatPseudocauchySequences 334
6.9 ConstructingCanonicalImmediateExtensions 335
7 Differential-HenselianFields 340
7.1 PreliminariesonDifferential-Henselianity 341
7.2 MaximalityandDifferential-Henselianity 345
7.3 Differential-HenselConfigurations 351
7.4 MaximalImmediateExtensionsintheMonotoneCase 353
CONTENTS ix
7.5 TheCaseofFewConstants 356
7.6 Differential-HenselianityinSeveralVariables 359
8 Differential-HenselianFieldswithManyConstants 365
8.1 AngularComponents 367
8.2 EquivalenceoverSubstructures 369
8.3 RelativeQuantifierElimination 374
8.4 AModelCompanion 377
9 AsymptoticFieldsandAsymptoticCouples 378
9.1 AsymptoticFieldsandTheirAsymptoticCouples 379
9.2 H-AsymptoticCouples 387
9.3 ApplicationtoDifferentialPolynomials 398
9.4 BasicFactsaboutAsymptoticFields 402
9.5 AlgebraicExtensionsofAsymptoticFields 409
9.6 ImmediateExtensionsofAsymptoticFields 413
9.7 DifferentialPolynomialsofOrderOne 416
9.8 ExtendingH-AsymptoticCouples 421
9.9 ClosedH-AsymptoticCouples 425
10 H-Fields 433
10.1 Pre-Differential-ValuedFields 433
10.2 AdjoiningIntegrals 439
10.3 TheDifferential-ValuedHull 443
10.4 AdjoiningExponentialIntegrals 445
10.5 H-FieldsandPre-H-Fields 451
10.6 LiouvilleClosedH-Fields 460
10.7 MiscellaneousFactsaboutAsymptoticFields 468
11 EventualQuantities,ImmediateExtensions,andSpecialCuts 474
11.1 EventualBehavior 474
11.2 NewtonDegreeandNewtonMultiplicity 482
11.3 UsingNewtonMultiplicityandNewtonWeight 487
11.4 ConstructingImmediateExtensions 492
11.5 SpecialCutsinH-AsymptoticFields 499
11.6 ThePropertyofλ-Freeness 505
11.7 BehavioroftheFunctionω 511
11.8 SomeSpecialDefinableSets 519
12 TriangularAutomorphisms 532
12.1 FilteredModulesandAlgebras 532
12.2 TriangularLinearMaps 541
12.3 TheLieAlgebraofanAlgebraicUnitriangularGroup 545
12.4 DerivationsontheRingofColumn-FiniteMatrices 548
12.5 IterationMatrices 552