Using algebraic geometry / David A. Cox, John Little, Donal O'Shea

Includes bibliographical references (pages 533-545) and index

Table Of Contents: Preface to the Second Edition Preface to the First Edition 1 Introduction 1 Polynomials and Ideals 2 Monomial Orders and Polynomial Division 3 Grobner Bases 4 Affine Varieties 2 Solving Polynomial Equations 1 Solving Polynomial Systems by Elimination 2 Finite-Dimensional Algebras 3 Grobner Basis Conversion 4 Solving Equations via Eigenvalues and Eigenvectors 5 Real Root Location and Isolation 3 Resultants 1 The Resultant of Two Polynomials 2 Multipolynomial Resultants 3 Properties of Resultants 4 Computing Resultants 5 Solving Equations via Resultants 6 Solving Equations via Eigenvalues and Eigenvectors 4 Computation in Local Rings 1 Local Rings 2 Multiplicities and Milnor Numbers 3 Term Orders and Division in Local Rings 4 Standard Bases in Local Rings 5 Applications of Standard Bases 5 Modules l Modules over Rings 2 Monomial Orders and Grobner Bases for Modules 3 Computing Syzygies 4 Modules over Local Rings 6 Free Resolutions 1 Presentations and Resolutions of Modules 2 Hilbert's Syzygy Theorem 3 Graded Resolutions 4 Hilbert Polynomials and Geometric Applications 7 Polytopes, Resultants, and Equations 1 Geometry of Polytopes 2 Sparse Resultants 3 Toric Varieties 4 Minkowski Sums and Mixed Volumes 5 Bernstein's Theorem 6 Computing Resultants and Solving Equations 8 Polyhedral Regions and Polynomials 1 Integer Programming 2 Integer Programming and Combinatorics 3 Multivariate Polynomial Splines 4 The Grobner Fan of an Ideal 5 The Grobner Walk 9 Algebraic Coding Theory 1 Finite Fields 2 Error-Correcting Codes 3 Cyclic Codes 4 Reed-Solomon Decoding Algorithms 10 The Berlekamp-Massey-Sakata Decoding Algorithm 1 Codes from Order Domains 2 The Overall Structure of the BMS Algorithm 3 The Details of the BMS Algorithm References Index