chosen in such a way that not all the diﬃculties, which are theoretically possible,
It therefore seems that Part C might be easier to read than the others, particularly
for those readers who are very familiar with computers and the concept of an al-
gorithm. Thus, such a reader could try to start with Part C to learn about the
experiments and to get acquainted with the theory. It is possible then to continue,
in a second step, with Parts A and B in order to get used to the theory in its
References and citations. When we refer to a definition, proposition, theorem,
etc., in the same chapter we simply rely on the corresponding numbering within
the chapter. Otherwise, we add the number of the chapter.
For the purpose of citation, the articles and books being used are encoded in the
manner specified by the bibliography. In addition, we mostly give the number of
the relevant section and subsection or the number of the definition, proposition,
theorem, etc. Normally, we do not mention page numbers.
Acknowledgments. I wish to acknowledge with gratitude my debt to Y. Tschinkel.
Most of the work described in this book, which is a shortened version of my Habil-
itation Thesis, was initiated by his numerous mathematical questions. During the
years he spent at Göttingen, he always shared his ideas in an extraordinarily gen-
I further wish to express my deep gratitude to my friend and colleague Andreas-
Stephan Elsenhans. He influenced this book in many ways, directly and indirectly.
It is no exaggeration to say that most of what I know about computer program-
ming I learned from him. The experiments, which are described in Part C, were
carried out together with him as a joint work. I am also indebted to Stephan
The computer part of the work described in this book was executed on the Sun Fire
V20z Servers of the Gauß Laboratory for Scientific Computing at the Göttingen
Mathematical Institute.The author is grateful to Y. Tschinkel for permission to use
these machines as well as to the system administrators for their support.