Asymptotic differential algebra and model theory of transseries / Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
- Author
- Aschenbrenner, Matthias, 1972-
- Published
- Princeton, New Jersey : Princeton University Press, 2017.
- Copyright Date
- ©2017
- Physical Description
- xxi, 849 pages ; 24 cm.
- Additional Creators
- Van den Dries, Lou and Hoeven, J. van der (Joris)
- Series
- Contents
- Machine generated contents note: 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 Differentia] Fields -- 6.1.Asymptotic Behavior of vυp -- 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.
- Subject(s)
- ISBN
- 9780691175423 hardcover ; alkaline paper
069117542X hardcover ; alkaline paper
9780691175430 paperback ; alkaline paper
0691175438 paperback ; alkaline paper - Bibliography Note
- Includes bibliographical references and index.
View MARC record | catkey: 20500854