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Ideals, Varieties, and Algorithms, Fifth Edition 2025
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Title: Ideals, Varieties, and Algorithms, Fifth Edition 2025 Subtitle: An Introduction to Computational Algebraic Geometry and Commutative Algebra Series: Undergraduate Texts in Mathematics Author: Cox, David A. / Little, John / O'Shea, Donal Publisher: Springer ISBN: 9783031918407 Cover: HARDCOVER Date: 2025年08月 DESCRIPTION This text covers topics in algebraic geometry and commutative algebra with careful attention to their practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry?the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz?there are chapters on polynomial and rational functions between varieties, robotics and geometric theorem proving, invariant theory of finite groups, projective algebraic geometry, dimension theory, and progress made over the last decades in computing Grobner bases. The fifth edition builds on the fourth edition in two main ways. First, a number of typographical errors, found by readers and by the authors since 2018, have been corrected. Second, new material on toric varieties, monomial curves, and other topics of current interest in algebraic geometry has been added. This enhances the opportunities for active learning through new examples, new exercises, and new projects in Appendix D, all supplemented by additional references. The book also includes updated computer algebra material in Appendix C. The book may be used for a first or second course in undergraduate abstract algebra and, with some augmentation perhaps, for beginning graduate courses in algebraic geometry or computational commutative algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple?, MathematicaR and SageMath, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used. TABLE OF 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 10 Additional Groebner Basis Algorithms
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