Asymptotic differential algebra and model theory of transseries /

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 logar...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Aschenbrenner, Matthias, 1972-
Otros Autores: Van den Dries, Lou, Hoeven, J. van der (Joris)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Princeton : Princeton University Press, 2017.
Colección:Annals of mathematics studies ; no. 195.
Temas:
Series, Arithmetic.
Divergent series.
Asymptotic expansions.
Differential algebra.
Séries arithmétiques.
Séries divergentes.
Développements asymptotiques.
Algèbre différentielle.
arithmetic progressions.
MATHEMATICS > Algebra > Intermediate.
MATHEMATICS > Algebra > Abstract.
Asymptotic expansions
Differential algebra
Divergent series
Series, Arithmetic
Equalizer Theorem.
H-asymptotic couple.
H-asymptotic field.
H-field.
Hahn Embedding Theorem.
Hahn space.
Johnson's Theorem.
Krull's Principal Ideal Theorem.
Kähler differentials.
Liouville closed H-field.
Liouville closure.
Newton degree.
Newton diagram.
Newton multiplicity.
Newton tree.
Newton weight.
Newton-Liouville closure.
Riccati transform.
Scanlon's extension.
Zariski topology.
algebraic differential equation.
algebraic extension.
angular component map.
asymptotic couple.
asymptotic differential algebra.
asymptotic field.
asymptotic relation.
asymptotics.
closed H-asymptotic couple.
closure properties.
coarsening.
commutative algebra.
commutative ring.
compositional conjugation.
constant.
continuity.
d-henselian.
d-henselianity.
decomposition.
derivation.
differential field extension.
differential field.
differential module.
differential polynomial.
differential-hensel.
differential-henselian field.
differential-henselianity.
differential-valued extension.
differentially closed field.
dominant part.
equivalence.
eventual quantities.
exponential integral.
extension.
filtered module.
gaussian extension.
grid-based transseries.
henselian valued field.
homogeneous differential polynomial.
immediate extension.
integral.
integrally closed domain.
linear differential equation.
linear differential operator.
linear differential polynomial.
mathematics.
maximal immediate extension.
model companion.
monotonicity.
noetherian ring.
ordered abelian group.
ordered differential field.
ordered set.
pre-differential-valued field.
pseudocauchy sequence.
pseudoconvergence.
quantifier elimination.
rational asymptotic integration.
regular local ring.
residue field.
simple differential ring.
small derivation.
special cut.
specialization.
substructure.
transseries.
triangular automorphism.
triangular derivation.
valuation topology.
valuation.
value group.
valued abelian group.
valued differential field.
valued field.
valued vector space.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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