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Author Aschenbrenner, Matthias, 1972-

Title Asymptotic differential algebra and model theory of transseries / Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
Published Princeton : Princeton University Press, 2017
Online access available from:
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Description 1 online resource
Series Annals of mathematics studies ; number 195
Annals of mathematics studies ; number 195
Contents 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 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
Bibliography Includes bibliographical references and index
Notes Print version record
Subject Asymptotic expansions.
Differential algebra.
Divergent series.
Series, Arithmetic.
Form Electronic book
Author Hoeven, J. van der (Joris)
Van den Dries, Lou.
ISBN 1400885418 (electronic bk.)
9781400885411 (electronic bk.)