Computational Methods in Commutative Algebra and Algebraic Geometry

This ACM volume in computational algebra deals with methods and techniques to tackle problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. It relates discoveries by a growing, interdisciplinary, group of researchers in the past decade. It highlights the use of advanced techniques to bring down the cost of computation. The book includes concrete algorithms written in MACAULAY. It is intended for advanced students and researchers with interests both in algebra and computation. Many parts of it can be read by anyone with a basic abstract algebra course. TOC:Fundamental Algorithms.- Toolkit.- Principles of Primary Decomposition.- Computing in Artin Algebras.- Nullstellensätze.- Integral Closure.- Ideal Transforms and Rings of Invariants.- Computation of Cohomology (by David Eisenbud).- Degrees of Complexity of a Graded Module.- Appendix A. A Primer on Commutative Algebra.- Appendix B. Hilbert Functions (by Jürgen Herzog).- Appendix C. Using Macaulay 2 (by David Eisenbud, Daniel Grayson and Michael Stillman.- Bibliography.- Index