Using Algebraic Geometry

Front Cover
Springer Science & Business Media, Mar 30, 2006 - Mathematics - 575 pages
In recent years, the discovery of new algorithms for dealing with polynomial equations, coupled with their implementation on fast inexpensive computers, has sparked a minor revolution in the study and practice of algebraic geometry. These algorithmic methods have also given rise to some exciting new applications of algebraic geometry. This book illustrates the many uses of algebraic geometry, highlighting some of the more recent applications of Gröbner bases and resultants. In order to do this, the authors provide an introduction to some algebraic objects and techniques which are more advanced than one typically encounters in a first course, but nonetheless of great utility. The book is written for nonspecialists and for readers with a diverse range of backgrounds. It assumes knowledge of the material covered in a standard undergraduate course in abstract algebra, and it would help to have some previous exposure to Gröbner bases. The book does not assume the reader is familiar with more advanced concepts such as modules. For this new edition the authors added two new sections and a new chapter, updated the references and made numerous minor improvements throughout the text.
 

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Contents

Preface to the Second Edition
1
Solving Polynomial Equations
26
Resultants
77
Computation in Local Rings
137
2 Multiplicities and Milnor Numbers
145
3 Term Orders and Division in Local Rings
158
4 Standard Bases in Local Rings
174
5 Applications of Standard Bases
180
Free Resolutions
247
Polytopes Resultants and Equations
305
Polyhedral Regions and Polynomials
376
Algebraic Coding Theory
451
The BerlekampMasseySakata Decoding Algorithm
494
References
533
Index
547
Copyright

Modules
189

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