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Asymptotic Differential Algebra and Model Theory of Transseries

Matthias Aschenbrenner, Lou van den Dries and Joris van der Hoeven
Publisher: 
Princeton University Press
Publication Date: 
2017
Number of Pages: 
880
Format: 
Paperback
Series: 
Annals of Mathematics Studies
Price: 
75.00
ISBN: 
9780691175430
Category: 
Monograph
We do not plan to review this book.

Preface xiii
Conventions and Notations xv
Leitfaden xvii
Dramatis Personæ xix
Introduction and Overview 1
A Differential Field with No Escape 1
Strategy and Main Results 10
Organization 21
The Next Volume 24
Future Challenges 25
A Historical Note on Transseries 26
1 Some Commutative Algebra 29
1.1 The Zariski Topology and Noetherianity 29
1.2 Rings and Modules of Finite Length 36
1.3 Integral Extensions and Integrally Closed Domains 39
1.4 Local Rings 43
1.5 Krull's Principal Ideal Theorem 50
1.6 Regular Local Rings 52
1.7 Modules and Derivations 55
1.8 Differentials 59
1.9 Derivations on Field Extensions 67
2 Valued Abelian Groups 70
2.1 Ordered Sets 70
2.2 Valued Abelian Groups 73
2.3 Valued Vector Spaces 89
2.4 Ordered Abelian Groups 98
3 Valued Fields 110
3.1 Valuations on Fields 110
3.2 Pseudoconvergence in Valued Fields 126
3.3 Henselian Valued Fields 136
3.4 Decomposing Valuations 157
3.5 Valued Ordered Fields 171
3.6 Some Model Theory of Valued Fields 179
3.7 The Newton Tree of a Polynomial over a Valued Field 186
4 Differential Polynomials 199
4.1 Differential Fields and Differential Polynomials 199
4.2 Decompositions of Differential Polynomials 209
4.3 Operations on Differential Polynomials 214
4.4 Valued Differential Fields and Continuity 221
4.5 The Gaussian Valuation 227
4.6 Differential Rings 231
4.7 Differentially Closed Fields 237
5 Linear Differential Polynomials 241
5.1 Linear Differential Operators 241
5.2 Second-Order Linear Differential Operators 258
5.3 Diagonalization of Matrices 264
5.4 Systems of Linear Differential Equations 270
5.5 Differential Modules 276
5.6 Linear Differential Operators in the Presence of a Valuation 285
5.7 Compositional Conjugation 290
5.8 The Riccati Transform 298
5.9 Johnson's Theorem 303
6 Valued Differential Fields 310
6.1 Asymptotic Behavior of vP 311
6.2 Algebraic Extensions 314
6.3 Residue Extensions 316
6.4 The Valuation Induced on the Value Group 320
6.5 Asymptotic Couples 322
6.6 Dominant Part 325
6.7 The Equalizer Theorem 329
6.8 Evaluation at Pseudocauchy Sequences 334
6.9 Constructing Canonical Immediate Extensions 335
7 Differential-Henselian Fields 340
7.1 Preliminaries on Differential-Henselianity 341
7.2 Maximality and Differential-Henselianity 345
7.3 Differential-Hensel Configurations 351
7.4 Maximal Immediate Extensions in the Monotone Case 353
7.5 The Case of Few Constants 356
7.6 Differential-Henselianity in Several Variables 359
8 Differential-Henselian Fields with Many Constants 365
8.1 Angular Components 367
8.2 Equivalence over Substructures 369
8.3 Relative Quantifier Elimination 374
8.4 A Model Companion 377
9 Asymptotic Fields and Asymptotic Couples 378
9.1 Asymptotic Fields and Their Asymptotic Couples 379
9.2 H-Asymptotic Couples 387
9.3 Application to Differential Polynomials 398
9.4 Basic Facts about Asymptotic Fields 402
9.5 Algebraic Extensions of Asymptotic Fields 409
9.6 Immediate Extensions of Asymptotic Fields 413
9.7 Differential Polynomials of Order One 416
9.8 Extending H-Asymptotic Couples 421
9.9 Closed H-Asymptotic Couples 425
10 H-Fields 433
10.1 Pre-Differential-Valued Fields 433
10.2 Adjoining Integrals 439
10.3 The Differential-Valued Hull 443
10.4 Adjoining Exponential Integrals 445
10.5 H-Fields and Pre-H-Fields 451
10.6 Liouville Closed H-Fields 460
10.7 Miscellaneous Facts about Asymptotic Fields 468
11 Eventual Quantities, Immediate Extensions, and Special Cuts 474
11.1 Eventual Behavior 474
11.2 Newton Degree and Newton Multiplicity 482
11.3 Using Newton Multiplicity and Newton Weight 487
11.4 Constructing Immediate Extensions 492
11.5 Special Cuts in H-Asymptotic Fields 499
11.6 The Property of l-Freeness 505
11.7 Behavior of the Function ! 511
11.8 Some Special Definable Sets 519
12 Triangular Automorphisms 532
12.1 Filtered Modules and Algebras 532
12.2 Triangular Linear Maps 541
12.3 The Lie Algebra of an Algebraic Unitriangular Group 545
12.4 Derivations on the Ring of Column-Finite Matrices 548
12.5 Iteration Matrices 552
12.6 Riordan Matrices 563
12.7 Derivations on Polynomial Rings 568
12.8 Application to Differential Polynomials 579
13 The Newton Polynomial 585
13.1 Revisiting the Dominant Part 586
13.2 Elementary Properties of the Newton Polynomial 593
13.3 The Shape of the Newton Polynomial 598
13.4 Realizing Cuts in the Value Group 606
13.5 Eventual Equalizers 610
13.6 Further Consequences of w-Freeness 615
13.7 Further Consequences of l-Freeness 622
13.8 Asymptotic Equations 628
13.9 Some Special H-Fields 635
14 Newtonian Differential Fields 640
14.1 Relation to Differential-Henselianity 641
14.2 Cases of Low Complexity 645
14.3 Solving Quasilinear Equations 651
14.4 Unravelers 657
14.5 Newtonization 665
15 Newtonianity of Directed Unions 671
15.1 Finitely Many Exceptional Values 671
15.2 Integration and the Extension K(x) 672
15.3 Approximating Zeros of Differential Polynomials 673
15.4 Proof of Newtonianity 676
16 Quantifier Elimination 678
16.1 Extensions Controlled by Asymptotic Couples 680
16.2 Model Completeness 685
16.3 LW-Cuts and LW-Fields 688
16.4 Embedding Pre-LW-Fields into w-Free LW-Fields 697
16.5 The Language of LW-Fields 701
16.6 Elimination of Quantifiers with Applications 704
A Transseries 712
B Basic Model Theory 724
B.1 Structures and Their Definable Sets 724
B.2 Languages 729
B.3 Variables and Terms 734
B.4 Formulas 738
B.5 Elementary Equivalence and Elementary Substructures 744
B.6 Models and the Compactness Theorem 749
B.7 Ultraproducts and Proof of the Compactness Theorem 755
B.8 Some Uses of Compactness 759
B.9 Types and Saturated Structures 763
B.10 Model Completeness 767
B.11 Quantifier Elimination 771
B.12 Application to Algebraically Closed and Real Closed Fields 776
B.13 Structures without the Independence Property 782
Bibliography 787
List of Symbols 817
Index 833

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