Transseries and Real Differential Algebra

Joris van der Hoeven (auth.)9783540355908, 9783540355915, 3540355901

Transseries are formal objects constructed from an infinitely large variable x and the reals using infinite summation, exponentiation and logarithm. They are suitable for modeling “strongly monotonic” or “tame” asymptotic solutions to differential equations and find their origin in at least three different areas of mathematics: analysis, model theory and computer algebra. They play a crucial role in Écalle’s proof of Dulac’s conjecture, which is closely related to Hilbert’s 16th problem. The aim of the present book is to give a detailed and self-contained exposition of the theory of transseries, in the hope of making it more accessible to non-specialists.

Table of contents :
Front Matter….Pages I-XXII
Orderings….Pages 11-32
Grid-based series….Pages 33-55
The Newton polygon method….Pages 57-77
Transseries….Pages 79-96
Operations on transseries….Pages 97-113
Grid-based operators….Pages 115-133
Linear differential equations….Pages 135-164
Algebraic differential equations….Pages 165-200
The intermediate value theorem….Pages 201-233
Back Matter….Pages 235-259