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Author Cox, David A.

Title Ideals, varieties, and algorithms : an introduction to computational algebraic geometry and commutative algebra : with 91 illustrations / David Cox, John Little, Donal O'Shea.

Published New York : Springer, [1997]


Location Call No. Status
 UniM Bund  516.35 COX {Bund89 20200519}    AVAILABLE
Edition 2nd ed.
Physical description xiii, 536 pages : illustrations ; 25 cm.
Series Undergraduate texts in mathematics.
Undergraduate texts in mathematics.
Bibliography Includes bibliographical references (pages 523-526) and index.
Contents 1. Geometry, Algebra, and Algorithms -- 2. Groebner Bases -- 3. Elimination Theory -- 4. The Algebra-Geometry Dictionary -- 5. Polynomial and Rational Functions on a Variety -- 6. Robotics and Automatic Geometric Theorem Proving -- 7. Invariant Theory of Finite Groups -- 8. Projective Algebraic Geometry -- 9. The Dimension of a Variety -- App. A. Some Concepts from Algebra -- App. B. Pseudocode -- App. C. Computer Algebra Systems -- App. D. Independent Projects.
Summary Algebraic geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.
The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960s. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have led to some interesting applications - for example, in robotics and in geometric theorem proving.
Other author Little, John B.
O'Shea, Donal.
Subject Geometry, Algebraic -- Data processing.
Commutative algebra -- Data processing.
ISBN 0387946802 (alk. paper)