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 Gr"obner 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 Gr"obner bases. The book does not assume the reader is familiar with more advanced concepts such as modules. |
Contents
| 1 | |
Solving Polynomial Equations | 24 |
Resultants | 71 |
Computation in Local Rings | 130 |
Modules | 179 |
Free Resolutions | 234 |
Polytopes Resultants and Equations | 290 |
Integer Programming Combinatorics | 359 |
Algebraic Coding Theory | 407 |
| 468 | |
| 477 | |
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Common terms and phrases
A₁ affine algebra Buchberger's Chapter codewords coefficients columns compute consider coordinates corresponding coset curve cyclic code defined Definition denote determinant dimension division algorithm eigenvalues eigenvectors elements entries equations exact sequence example Exercise 11 Exercise 9 f₁ finite formula free module free resolution geometric given gives graded resolution Gröbner basis Hence Hilbert function Hilbert polynomial Hint homogeneous polynomials homomorphism implies irreducible isomorphism leading terms Lemma lex order linear code local ring LT(I M₁ method minimal Minkowski sum mixed subdivision mixed volume monomial order multiplication nontrivial solution nonzero obtain P₁ points polynomial ring polytope presentation matrix proof of Theorem Proposition prove quotient ring R-module Res(f roots Show sparse resultant splines standard basis submodule subset subspace syzygies theory total degree variables variety vector space zero zero-dimensional ideal