Asymptotic Differential Algebra and Model Theory of Transseries / Joris van der Hoeven, Matthias Aschenbrenner, Lou van den Dries.

Format:

Book

Author/Creator:

Aschenbrenner, Matthias, author.

Van den Dries, Lou, author.

van der Hoeven, Joris, author.

Series:

Annals of mathematics studies ; no. 195.

Annals of Mathematics Studies ; 358

Language:

English

Subjects (All):

Series, Arithmetic.

Divergent series.

Asymptotic expansions.

Differential algebra.

Physical Description:

1 online resource (874 pages) : illustrations.

Place of Publication:

Princeton, NJ : Princeton University Press, [2017]

Language Note:

In English.

Summary:

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.

Contents:

Frontmatter

Contents

Preface

Conventions and Notations

Leitfaden

Dramatis Personæ

Introduction and Overview

Chapter One. Some Commutative Algebra

Chapter Two. Valued Abelian Groups

Chapter Three. Valued Fields

Chapter Four. Differential Polynomials

Chapter Five. Linear Differential Polynomials

Chapter Six. Valued Differential Fields

Chapter Seven. Differential-Henselian Fields

Chapter Eight. Differential-Henselian Fields with Many Constants

Chapter Nine. Asymptotic Fields and Asymptotic Couples

Chapter Ten. H-Fields

Chapter Eleven. Eventual Quantities, Immediate Extensions, and Special Cuts

Chapter Twelve. Triangular Automorphisms

Chapter Thirteen. The Newton Polynomial

Chapter Fourteen. Newtonian Differential Fields

Chapter Fifteen. Newtonianity of Directed Unions

Chapter Sixteen. Quantifier Elimination

Appendix A. Transseries

Appendix B. Basic Model Theory

Bibliography

List of Symbols

Index

Notes:

Previously issued in print: 2017.

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

ISBN:

9781400885411

1400885418

OCLC:

984643717