Asymptotic Differential Algebra and Model Theory of Transseries

Cover
Title
Copyright
Contents
Preface
Conventions and Notations
Leitfaden
Dramatis Personæ
Introduction and Overview
	A Differential Field with No Escape
	Strategy and Main Results
	Organization
	The Next Volume
	Future Challenges
	A Historical Note on Transseries
1 Some Commutative Algebra
	1.1 The Zariski Topology and Noetherianity
	1.2 Rings and Modules of Finite Length
	1.3 Integral Extensions and Integrally Closed Domains
	1.4 Local Rings
	1.5 Krull’s Principal Ideal Theorem
	1.6 Regular Local Rings
	1.7 Modules and Derivations
	1.8 Differentials
	1.9 Derivations on Field Extensions
2 Valued Abelian Groups
	2.1 Ordered Sets
	2.2 Valued Abelian Groups
	2.3 Valued Vector Spaces
	2.4 Ordered Abelian Groups
3 Valued Fields
	3.1 Valuations on Fields
	3.2 Pseudoconvergence in Valued Fields
	3.3 Henselian Valued Fields
	3.4 Decomposing Valuations
	3.5 Valued Ordered Fields
	3.6 Some Model Theory of Valued Fields
	3.7 The Newton Tree of a Polynomial over a Valued Field
4 Differential Polynomials
	4.1 Differential Fields and Differential Polynomials
	4.2 Decompositions of Differential Polynomials
	4.3 Operations on Differential Polynomials
	4.4 Valued Differential Fields and Continuity
	4.5 The Gaussian Valuation
	4.6 Differential Rings
	4.7 Differentially Closed Fields
5 Linear Differential Polynomials
	5.1 Linear Differential Operators
	5.2 Second-Order Linear Differential Operators
	5.3 Diagonalization of Matrices
	5.4 Systems of Linear Differential Equations
	5.5 Differential Modules
	5.6 Linear Differential Operators in the Presence of a Valuation
	5.7 Compositional Conjugation
	5.8 The Riccati Transform
	5.9 Johnson’s Theorem
6 Valued Differential Fields
	6.1 Asymptotic Behavior of vP
	6.2 Algebraic Extensions
	6.3 Residue Extensions
	6.4 The Valuation Induced on the Value Group
	6.5 Asymptotic Couples
	6.6 Dominant Part
	6.7 The Equalizer Theorem
	6.8 Evaluation at Pseudocauchy Sequences
	6.9 Constructing Canonical Immediate Extensions
7 Differential-Henselian Fields
	7.1 Preliminaries on Differential-Henselianity
	7.2 Maximality and Differential-Henselianity
	7.3 Differential-Hensel Configurations
	7.4 Maximal Immediate Extensions in the Monotone Case
	7.5 The Case of Few Constants
	7.6 Differential-Henselianity in Several Variables
8 Differential-Henselian Fields with Many Constants
	8.1 Angular Components
	8.2 Equivalence over Substructures
	8.3 Relative Quantifier Elimination
	8.4 A Model Companion
9 Asymptotic Fields and Asymptotic Couples
	9.1 Asymptotic Fields and Their Asymptotic Couples
	9.2 H-Asymptotic Couples
	9.3 Application to Differential Polynomials
	9.4 Basic Facts about Asymptotic Fields
	9.5 Algebraic Extensions of Asymptotic Fields
	9.6 Immediate Extensions of Asymptotic Fields
	9.7 Differential Polynomials of Order One
	9.8 Extending H-Asymptotic Couples
	9.9 Closed H-Asymptotic Couples
10 H-Fields
	10.1 Pre-Differential-Valued Fields
	10.2 Adjoining Integrals
	10.3 The Differential-Valued Hull
	10.4 Adjoining Exponential Integrals
	10.5 H-Fields and Pre-H-Fields
	10.6 Liouville Closed H-Fields
	10.7 Miscellaneous Facts about Asymptotic Fields
11 Eventual Quantities, Immediate Extensions, and Special Cuts
	11.1 Eventual Behavior
	11.2 Newton Degree and Newton Multiplicity
	11.3 Using Newton Multiplicity and Newton Weight
	11.4 Constructing Immediate Extensions
	11.5 Special Cuts in H-Asymptotic Fields
	11.6 The Property of λ-Freeness
	11.7 Behavior of the Function ω
	11.8 Some Special Definable Sets
12 Triangular Automorphisms
	12.1 Filtered Modules and Algebras
	12.2 Triangular Linear Maps
	12.3 The Lie Algebra of an Algebraic Unitriangular Group
	12.4 Derivations on the Ring of Column-Finite Matrices
	12.5 Iteration Matrices
	12.6 Riordan Matrices
	12.7 Derivations on Polynomial Rings
	12.8 Application to Differential Polynomials
13 The Newton Polynomial
	13.1 Revisiting the Dominant Part
	13.2 Elementary Properties of the Newton Polynomial
	13.3 The Shape of the Newton Polynomial
	13.4 Realizing Cuts in the Value Group
	13.5 Eventual Equalizers
	13.6 Further Consequences of ω-Freeness
	13.7 Further Consequences of λ-Freeness
	13.8 Asymptotic Equations
	13.9 Some Special H-Fields
14 Newtonian Differential Fields
	14.1 Relation to Differential-Henselianity
	14.2 Cases of Low Complexity
	14.3 Solving Quasilinear Equations
	14.4 Unravelers
	14.5 Newtonization
15 Newtonianity of Directed Unions
	15.1 Finitely Many Exceptional Values
	15.2 Integration and the Extension K(x)
	15.3 Approximating Zeros of Differential Polynomials
	15.4 Proof of Newtonianity
16 Quantifier Elimination
	16.1 Extensions Controlled by Asymptotic Couples
	16.2 Model Completeness
	16.3 ΛΩ-Cuts and ΛΩ-Fields
	16.4 Embedding Pre-ΛΩ-Fields into ω-Free ΛΩ-Fields
	16.5 The Language of ΛΩ-Fields
	16.6 Elimination of Quantifiers with Applications
A Transseries
B Basic Model Theory
	B.1 Structures and Their Definable Sets
	B.2 Languages
	B.3 Variables and Terms
	B.4 Formulas
	B.5 Elementary Equivalence and Elementary Substructures
	B.6 Models and the Compactness Theorem
	B.7 Ultraproducts and Proof of the Compactness Theorem
	B.8 Some Uses of Compactness
	B.9 Types and Saturated Structures
	B.10 Model Completeness
	B.11 Quantifier Elimination
	B.12 Application to Algebraically Closed and Real Closed Fields
	B.13 Structures without the Independence Property
Bibliography
List of Symbols
Index