Cambridge University Press: Cambridge, 1998, viii+552 p (London Mathematical Society Lecture Note Series 251.

ISBN 0 521 63298 6 (paperback)

**Review by Hans Sterk**

RIACA/Department of Mathematics and Computing Science

Eindhoven University of Technology, The Netherlands

Email: `sterk@win.tue.nl`

Gröbner bases were introduced 33 years ago by Bruno Buchberger, but their true impact came several years later. I guess that only in the 80s things really took off and people in commutative algebra and algebraic geometry became aware of the usefulness of the concept of a Gröbner basis. With the availability now of computer algebra packages to do computations with Gröbner bases, they have become serious business in research as well as in (undergraduate) teaching. Today, it is a well-established concept at the cross roads of various classical and modern branches of mathematics, e.g., commutative algebra and computer algebra.

To celebrate the fact that Gröbner bases have been around now for a third of a century, a meeting was organized at RISC-Linz - Buchberger is chairman there - at the end of a special year on Gröbner bases. The book contains the contributions for this meeting of many people involved in one way or another with Gröbner bases.

The book consists of two parts, called `Tutorials' and `Research Papers',
and an appendix. The appendix comes last, but goes back to the beginning
of the story on Gröbner bases: it contains the English translation of the journal version of Buchberger's paper *Ein algoritmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungsystems*
(An algorithmic Criterion for the solvability of a system of algebraic equations) in Aeq. Math. 4 (1970).

The tutorials show the diversity of fields in which Gröbner bases have come to play a role. Most titles in this part of the book read like `Gröbner bases and ...'. Here is a list of the subjects touched upon: symbolic summation and integration, invariant theory, generic initial ideals, algebraic geometry, integer programming, numerical analysis, statistics, coding theory, Janet bases for symmetry groups, partial differential equations, hypergeometric functions, noncommutative Gröbner basis theory, geometric theorem proving and discovering.

The part `Research Papers' brings together seventeen papers with new results, illustrating the constant activity around Gröbner bases and the diversity of subjects where they have become a vital tool or have given rise to new variations. To name a few of the subjects: Gröbner bases with respect to different term orders (Amrhein, Gloor), Gröbner bases in the computer algebra system CoCoA (Capani, Niesi), Gröbner bases in rings of differential operators (Insa, Pauer), Gröbner bases and Design of Experiments (Robbiano, Rogantin), Gröbner bases and the inversion of birational maps (Schicho). It also contains an `old' ('83) result on the resolution of ideals, which Mora decided to present in this volume `to show a piece of research in those times when the researchers on Gröbner could have been counted on the fingers of two hands...' (Mora's words).

To conclude, I think the book serves its purpose to give a quick overview of the state of the art of Gröbner bases; it is a valuable contribution to the literature.