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).

• Preface xiii Conventions and Notations xv Leitfaden xvii Dramatis Personae 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

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).

• Preface xiii Conventions and Notations xv Leitfaden xvii Dramatis Personae 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

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).

• Preface xiii Conventions and Notations xv Leitfaden xvii Dramatis Personae 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