Gröbner bases were introduced in 1965 by the first editor (Buchberger) in his Ph.D. thesis. It took some years before the concept became known to the research communities in Mathematics and Theoretical Computer Science, but about ten years later many other research groups around the world started working on or using this concept. The theory of Gröbner bases has become an important subarea in computer algebra, it is included in all the major program systems of symbolic computation, and it is being fruitfully applied in a variety of seemingly unrelated research areas.

We at RISC-Linz wanted to celebrate the fact that Gröbner bases have been around now for a third of a century, and in doing so provide a snapshot of the state of the art. For this reason we organized a Special Year on Gröbner Bases, culminating in an intensive course for young researchers (January 1998) and in the conference “33 Years of Gröbner Bases”, held at RISC-Linz in February 2–4, 1998. The conference included both tutorials on various aspects of Gröbner bases and their application, and the presentation of original research papers on new developments in the theory of Gröbner bases.

The present book is the outcome of these activities. It contains a short introduction to the theory of Gröbner bases, tutorial papers on the interaction between Gröbner bases and various other mathematical theories, and the original research papers presented at the conference.