Using Algebraic GeometryIn 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. |
Contents
| 1 | |
5 Real Root Location and Isolation | 69 |
Resultants | 77 |
Computation in Local Rings | 137 |
Modules | 189 |
Free Resolutions | 247 |
Polytopes Resultants and Equations | 305 |
2 Sparse Resultants | 320 |
Polyhedral Regions and Polynomials | 376 |
2 Integer Programming and Combinatorics | 392 |
Algebraic Coding Theory | 451 |
The BerlekampMasseySakata Decoding Algorithm | 494 |
| 543 | |
| 554 | |
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Common terms and phrases
affine algebra basis G Chapter codewords coefficients columns compute consider corresponding cosets cyclic code decoding defined Definition denote determinant dimension eigenvalues eigenvectors elements entries equations example Exercise 11 Exercise 9 f₁ finite formula free module free resolution geometric given gives graded resolution grevlex Gröbner basis Hence Hilbert function Hilbert polynomial Hint homogeneous polynomials homomorphism implies integer isomorphism leading terms Lemma lex order linear linear code local ring M₁ method minimal Minkowski sum mixed subdivision mixed volume monomial order multiplication Newton polytopes nontrivial solution nonzero obtain order domain points polynomial ring polytope presentation matrix Proposition prove quotient ring R-module Res(f respect roots Show sparse resultant splines standard basis submodule subset syzygies total degree variables variety vector space zero zero-dimensional ideal