Asymptotic Differential Algebra and Model Theory of Transseries : : (AMS-195) / / Matthias Aschenbrenner, Joris van der Hoeven, Lou van den Dries.

• Superior document: Title is part of eBook package: De Gruyter EBOOK PACKAGE COMPLETE 2017
• VerfasserIn: Aschenbrenner, Matthias,; van den Dries, Lou,; van der Hoeven, Joris,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2017]; ©2017
• Year of Publication: 2017
• Language: English
• Series: Annals of Mathematics Studies ; 195
• Physical Description: 1 online resource (880 p.) :; 12 line illus.
• Other title: 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
• Format: Mode of access: Internet via World Wide Web.
• ISBN: 9781400885411; 9783110540550; 9783110548204; 9783110494914; 9783110543322
• DOI: 10.1515/9781400885411?locatt=mode:legacy
• Access: restricted access
• Hierarchical level: Monograph
• Statement of Responsibility: Matthias Aschenbrenner, Joris van der Hoeven, Lou van den Dries.

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.