Using Algebraic Geometry
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 Grobner bases. The book does not assume the reader is familiar with more advanced concepts such as modules.
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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.
CONTENTS:Preface 1 Introduction Polynomials and Ideals Monomial Orders and Polynomial Division Grobner Bases Affine Varieties 2 Solving Polynomial Equations Solving Polynomial Systems by Elimination Finite-Dimensional Algebras Grobner Basis Conversion Solving Equations via Eigenvalues Real Root Location and Isolation 3. Resultants The Resultant of Two Polynomials Multipolynomial Resultants Properties of Resultants Computing Resultants Solving Equations via Resultants Solving Equations via Eigenvalues 4. Computation in Local Rings Local Rings Multiplicities and Milnor Numbers Term Orders and Division in Local Rings Standard Bases in Local Rings 5. Modules Modules over Rings Monomial Orders and Grobner Bases for Modules Computing Syzygies Modules over Local Rings 6. Free Resolutions Presentations and Resolutions of Modules Hilbert's Syzygy Theorem Graded Resolutions Hilbert Polynomials and Geometric Applications 7. Polytopes, Resultants and Equations Geometry of Polytopes Sparse Resultants Toric Varieties Minkowski Sums and Mixed Volumes Bernstein's Theorem Computing Resultants and Solving Equations 8. Integer Programming, Combinatorics and Splines Integer Programming Integer Programming and Combinatorics Multivariate Polynomial Splines 9. Algebraic Coding Theory Finite Fields Error-Correcting Codes Cyclic Codes Reed-Solomon Decoding Algorithms Codes from Algebraic Geometry References Index
