Gröbner Bases

One can approach the study of algebra from two different points of view: axiomatically or algorithmically. While the former has been the dominant approach since the nineteenth century, both are equally venerable (and both take their names from the same medieval Arabic treatise). Recently, however, the explosive development of computer power has led to a renewed interest in algorithms and questions of computability and has established computer algebra as an independent field at the interface between mathematics and computer science. Gröbner bases, defined by Buchberger in 1965, and the Buchberger algorithm extend the euclidean algorithm for computing the greatest common divisor to polynomials in several variables. Algorithms using Gröbner bases lead to exact conclusions concerning the solutions of systems of nonlinear equations, such as the number of solutions or the dimension of the solution set, and the computation of the solutions to arbitrary precision. Such algorithms are implemented in most major computer algebra software systems. This book on Gröbner bases, assuming the mathematical background of an advanced undergraduate, will be both a reference manual for the working mathematician or computer algebraist as well as an introductory textbook.