Computational Methods in Commutative Algebra and Algebraic Geometry

Description

The interplay between computation and many areas of algebra is a natural phenomenon in view of the algorithmic character of the latter. The existence of inexpensive but powerful computational resources has enhanced these links by the opening up of many new areas of investigation in algebra. At the same time it made available the theoretical tools of this area of mathematics to help deal with problems of interest in physics, engineering and other disciplines. We aim here to discuss how certain devices that permit the rapid processing of polynomials and matrices make it possible to examine parts of two areas of algebra - commutative algebra and algebraic geometry - where those data structures play critical roles. Among the main tasks in computational algebra are the constructions of decompositions and closures of objects in the ring of polynomials. Among the former are finding primary decompositions and modules of syzygies, and among the latter, the computation of integral closures and of rings of invariants. As a rule, they are assisted by any a priori knowledge available. Another frequent task is to certify that a given object has a certain property. This book is an attempt to deal with these issues, despite the pace of development in the field (ring?). The material was drawn mostly from the published literature, both classical and recent, including conference proceedings on computer algebra that tended to focus on algebraic structures.