**Abstract**
In this thesis, we present the construction of fields with functions
which are faster than every iterated exponential function. This
introduction will describe what we mean by “construction”,
“faster than” and “exponential function”. By
doing this, we hope to give the reader a good idea of what he can expect
from this thesis, and we hope to provide a motivation for the presented
work. Moreover, this introduction will serve as a guide to help the
reader through the different parts of the thesis.

We start by explaining some basic concepts and by presenting the main
results. We go on to summarize what is known about super-exponential
functions. The third part of this introduction will motivate the given
construction. Then we will come to the “road map” of the
thesis: we give a short summary of each of the forthcoming chapters,
thus equipping a possible reader with an orientation guide. This will be
of particular interest since some chapters are rather technical, and
there is a real danger of losing the overview when working through the
unavoidable details. Finally, we list some of the notations used.

**Author:** Michael Schmeling

**Supervisor:** Jean-Pierre Ressayre

**Co-advisor:** Joris van der Hoeven

**View:** Pdf, BibTeX

**Historical notes**
A few words are in order to explain why I made Schmeling's
PhD. available on my website.

In 1998, Jean-Pierre Ressayre proposed me to
co-advise the PhD. work of Michael Schmeling,
who was then in his second year, and not making sufficient progress on
his initial subject. The plan was to find a new subject on which it
would be possible to obtain quick results and complete a thesis within
one year less than usual. The resulting strategy was to let him write
down and work out some of my ideas on generalized transseries with the
intention that he would gradually give this work a more personal turn.
As a consequence, a large part of Schmeling's
PhD. should really be regarded as the fruit of a
collaboration between us. Standard procedure dictates that we should
have co-authored one or two papers after completion of the thesis.
However, Michael Schemeling left for the private
sector, which prevented this from happening. Nevertheless, he proposed
to put the thesis on my personal website and advertise for it there.

It is difficult to spell out my own contributions to Schmeling's
PhD. in a satisfactory manner. Obviously,
Michael Schmeling did most of the hard work of
writing things down and working out many of the technical details. My
main personal contributions were to propose precise axioms for fields of
transseries with and without super-logarithmic functions, and to propose
the framework of strong linear algebra (see also my paper *“Operators
on generalized power series”*), derivations, and
compositions. This involvement was particularly important in the first
five chapters, which can be regarded as a continuation of the first two
chapters of my own PhD.; some of this material is also
present for the grid-based setting in my lecture notes.

Schmeling's thesis was actually part of a larger
project which fascinated me those times: developing a sufficiently rich
algebraic framework for expressing all tame (i.e. strongly
monotonic) solutions to systems of functional equations (involving
differentiation and composition). The profusion of technical subtleties,
combined with the lack of subsequent students to work on this subject,
discouraged me to complete this project. Nevertheless, this enterprise
still appears very worthwhile, especially in the light of connections
with Conway's theory of surreal numbers that we were not aware of at the
time.

It should also be pointed out that Ressayre's role
in the project ended up being purely administrative. Here one should
keep in mind that the French system requires the official
PhD. advisor to be “accredited to lead
research”. This means that one should have defended a second kind
of PhD., called “habilitation à diriger des
recherches”. I only did this six years after Schmeling's
PhD. defense. This explains why Ressayre
remained the official advisor, even though all the scientific
supervision was done by me. Unfortunately, my role as a supervisor has
not officially been acknowledged so far on the website of the University
Paris-VII. One official trace appears in Chapter 1 of the “activity report” that I wrote in 2000 for the CNRS.