Asymptotic Differential Algebra and Model Theory of Transseries

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

Overview

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems.

This self-contained book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton-Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

Product Details

ISBN-13:	9780691175430
Publisher:	Princeton University Press
Publication date:	06/06/2017
Series:	Annals of Mathematics Studies , #195
Pages:	880
Product dimensions:	5.90(w) x 9.20(h) x 1.80(d)

About the Author

Matthias Aschenbrenner is professor of mathematics at the University of California, Los Angeles. Lou van den Dries is professor of mathematics at the University of Illinois, Urbana-Champaign. Joris van der Hoeven is director of research at the French National Center for Scientific Research (CNRS).

Table of Contents

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