Includes bibliographical references and index.

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Cover; Title; Copyright; Contents; Preface; Conventions and Notations; Leitfaden; Dramatis Person�; 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 Extensions2 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 Continuity4.5 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 Extensions6.4 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 Components8.2 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.

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.

In English.

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