理想数、簇与算法 第2版（影印版）

　　We wrote this book to introduce undergraduates to some interesting ideas in algebraic geometry and commutative algebra. Until recently, these topics involved a lot of abstract mathematics and were only taught in graduate school. But in the 1960s, Buchberger and Hironaka discovered new algorithms for manipulating systems of polynomial equations. Fueled by the development of computers fast enough to run these algorithms, the last two decades have seen a minor revolution in commutative algebra. The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand, and has changed the practice of much research in algebraic geometry. This has also enhanced the importance of the subject for computer scientists and engineers, who have begun to use these techniques in a whole range of problems.
　　

preface to the first edition
preface to the second edition
1. geometry, algebra, and algorithms
1. polynomials and affine space
2. affine varieties
3. parametrizations of affine varieties
4. ideals
5. polynomials of one variable
2. groebner bases
1. introduction
2. orderings on the monomials in k[x1,....,xn]
3. a division algorithm in k[x1,....,xn]
4. monomial ideals and dickson's lemma
5. the hilbert basis theorem and groebner bases
6. properties of groebner bases
7. buchberger's algorithm
8. first applications of groebner bases
9. (optional) improvements on buchberger's algorithm
3. elimination theory
1. the elimination and extension theorems

　　We wrote this book to introduce undergraduates to some interesting ideas in algebraic geometry and commutative algebra. Until recently, these topics involved a lot of abstract mathematics and were only taught in graduate school. But in the 1960s, Buchberger and Hironaka discovered new algorithms for manipulating systems of polynomial equations. Fueled by the development of computers fast enough to run these algorithms, the last two decades have seen a minor revolution in commutative algebra. The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand, and has changed the practice of much research in algebraic geometry. This has also enhanced the importance of the subject for computer scientists and engineers, who have begun to use these techniques in a whole range of problems.
　　 It is our belief that the growing importance of these computational techniques warrants their introduction into the undergraduate (and graduate) mathematics curriculum. Many undergraduates enjoy the concrete, almos.t nineteenth-century, flavor that a computational emphasis brings to the subject. At the same time, one can do some substantial mathematics, including the Hilbert Basis Theorem, Elimination Theory, and the Nullstellensatz.
　　 The mathematical prerequisites of the book are modest: the students should have had a course in linear algebra and a course where they learned how to do proofs. Examples of the latter sort of course include discrete math and abstract algebra. It is important to note that abstract algebra is not a prerequisite. On the other hand, if all of the students have had abstract algebra, then certain parts of the course will go much more quickly.
　　 The book assumes that the students will have access to a computer algebra system. Appendix C describes the features of AXIOM, Maple, Mathematica, and REDUCE that are most relevant to the text. We do not assume any prior experience with a computer. However, many of the algorithms in the book are described in pseudocode, which may be unfamiliar to students with no background in programming. Appendix B contains a careful description of the pseudocode that we use in the text.
　　 In writing the book, we tried to structure the material so that thebook could be used in a variety of courses, and at a variety of different levels. For instance, the book could serve as a basis of a second course in undergraduate abstract algebra, but we think that it just as easily could provide a credible alternative to the first course. Although the book is aimed primarily at undergraduates, it could also be used in various graduate courses, with some supplements. In particular, beginning graduate courses in algebraic geometry or computational algebra may find the text useful. We hope, of course, that mathematicians and colleagues in other disciplines will enjoy reading the book as much as we enjoyed writing it.
　　 The first four chapters form the core of the book. It should be possible to cover them in a 14-week semester, and there may be some time left over at the end to explore other parts of the text. The follows chart explains the logical dependence of the chapters: See the table of contents for a description of what is covered in each chapter. As the chart indicates, there are a variety of ways to proceed after covering the first four chapters. Also, a two-semester course could be designed that covers the entire book. For instructors interested in having their students do an independent project, we have included a list of possible topics in Appendix D.
　　 It is a pleasure to thank the New England Consortium for Undergraduate Science Education (and its parent organization, the Pew Charitable Trusts) for providing the major funding for this work. The project would have been impossible without their support. Various aspects of our work were also aided by grants from IBM and the Sloan Foundation, the Alexander von Humboldt Foundation, the Department of Education's F1PSE program, the Howard Hughes Foundation, and the National Science Foundation. We are grateful for their help.
　　 We also wish to thank colleagues and students at Amherst College, George Mason University, Holy Cross College, Massachusetts Institute of Technology, Mount Holyoke College, Smith College, and the University of Massachusetts who participated in courses based on early versions of the manuscript. Their feedback improved the book considerably. Many other colleagues have contributed suggestions, and we thank you all.
　　 Corrections, comments and suggestions for improvement are welcome!
　　November 1991
　　David Cox
　　John Little
　　Donal O' Shea
　　In preparing a new edition of Ideals, Varieties, and Algorithms, our goal was to correct some of the omissions of the first edition while maintaining the readability and accessibility of the original. The majors changes in the second edition are as follows:
　　 Chapter 2: A better acknowledgement of Buchberger's contributions and an improved proof of the Buchberger Criterion in 6.
　　 Chapter 5: An improved bound on the number of solutions in 3 and a new 6 which completes the proof of the Closure Theorem begun in Chapter 3.
　　 Chapter 8: A complete proof of the Projection Extension Theorem in 5 and a new 7 which contains a proof of Bezout's Theorem.
　　 Appendix C: a new section on AXIOM and an update on what we say about Maple, Mathematica, and REDUCE.
　　Finally, we fixed some typographical errors, improved and clarified notation, and updated the bibliography by adding many new references.
　　 We also want to take this opportunity to acknowledge our debt to the many people who influenced us and helped us in the course of this project. In particular, we would like to thank: