Transseries and real differential algebra. (English) Zbl 1128.12008

van der Hoeven, Joris

Lecture Notes in Mathematics 1888. Berlin: Springer (ISBN 3-540-35590-1/pbk). xii, 255 p. (2006).

This book presents the field \(\mathbb T\) of transseries and some of its properties. A transseries can be described, speaking loosely, as a formal object constructed from the real numbers and an infinitely large variable \(x\) using infinite summation, exponentiation, and logarithm. In particular, \(\mathbb T\) is shown to be totally ordered, real closed, and stable under differentiation, integration, composition, and functional inversion. It also satisfies a differential intermediate value property: given a differential polynomial \(P\) over \(\mathbb T\) for which there exist transseries \(f < g\) with \(P(f)P(g) < 0\) then there is a transseries \(h \in \mathbb T\) with \(f < h < g\) for which \(P(h)=0\). This implies that any algebraic differential equation of odd degree over \(\mathbb T\) has a solution in \(\mathbb T\).
The statements in the preceding paragraph are essentially quotations from the author’s introduction. In fact the precise definition of \(\mathbb T\) comes in the middle of chapter 4 of this nine chapter volume, and the proof of the intermediate value property comes in the final section of the last chapter.
Chapters 1, 2, and 3 on orderings, grid-based series, and the Newton polygon method, lay the foundations for the definition of transseries in chapter 4. Chapters 5 and 6, on operations on transseries and grid–based operators, give the non–differential closure properties of \(\mathbb T\). Chapters 7, 8, and 9 then cover linear differential equations, algebraic differential equations, and the intermediate value theorem, for \(\mathbb T\).
The author intends the book for non-specialists, including graduate students, and to that end has made the volume self-contained and included exercises. The book is intended for mathematicians working in analysis, model theory, or computer algebra. Algebraists should also find interest in the algebraic properties of the field of transseries.

Reviewer: Andy R. Magid (Norman)

MSC:

12H05	Differential algebra
12-02	Research exposition (monographs, survey articles) pertaining to field theory
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
13N99	Differential algebra
34M37	Resurgence phenomena (MSC2000)
68W30	Symbolic computation and algebraic computation

Keywords:

differential field; infinite summation