Sinc Methods for Quadrature and Differential Equations (J. Lund and K. L. Bowers)

In algebraic geometry many natural problems, such as those arising in invariant theory, in the solution of polynomial systems of interest to scientists and engineers, and in the computation of syzygies of ideals, lead to computations that prove too tedious to pursue by hand. The widespread availability of powerful, relatively easy to use computers has spurred a return to these basic questions, and a rethinking of what reasonable problems are. Algorithms based on the work of Buchberger and many other mathematicians have led to a wide array of computational algebraic tools that are readily available on many computers. The very readable, excellent book being reviewed is an introduction to algebraic geometry emphasizing the algorithmic, computational viewpoint. This book will be valuable to many different constituencies. The core book consists of four chapters covering the correspondence between results on affine varieties and basic commutative algebra, a solid introduction to Grtibner bases, resultants, and elimination theory. The book then goes on to sample a number of more advanced topics, including basic material on the kinematics of planar robots, some invariant theory of finite groups, projective varieties, and dimension. All the chapters contain many very good examples and exercises. It will be easy to use this textbook as the basis for an innovative algebraic geometry course for undergraduate mathematics majors or for graduate students in a wide variety of subjects such as computer science and mechanical engineering. Mathematicians and scientists who are not specialists in computer algebra, and graduate students learning algebraic geometry by one of the many other approaches to the subject will also find this book useful and pleasant reading.