理想、簇与算法

Preface to the First EditionPreface to the Second EditionPreface to the Third Edition1 Geometry， Algebra， and Algoritlnns1. Polynomials and Affine Space2. Affine Varieties3. Parametrizations of Affine Varieties4. Ideals5. Polynomials of One Variable2. Groebner Bases2. Orderings on the Monomials in k1. Introduction3. A Division Algorithm in k4. Monomial Ideals and Dickson's Lemma5. The Hilbert Basis Theorem and Groebner Bases6. Properties of Groebner Bases7. Buchberger's Algorithm8. First Applications of Groebner Bases9. (Optional) Improvements on Buchberger's Algorithm3. Elimination Theory1. The Elimination and Extension Theorems2. The Geometry of Elimination3. Implicitization4. Singular Points and Envelopes5. Unique Factorization and Resultants6. Resultants and the Extension Theorem……3. Elimination Theory4. The Algebra-Geometry Dictionary5. Polynomial and Rational Functions on a Variety6. Robotics and Automatic Geometric Theorem Proving7. Invariant Theory of Finite Groups8. Projective Algebraic Geometry9. The Dimension of a VarietyAppendix A.Some Concepts From AlgebraAppendix B.PseudocodeAppendix C.Computer Algebra SystemsAppendix D.Independent ProjectsReferencesIndex

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 com-puter. 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 con-tains a careful descripLion of the pseudocode that we use in the text.