Corps de transséries
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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

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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.