本卷收录了吴文俊的Mathematics Mechanization: Mechanical Geometry Theorem-Proving, Mechanical Geometry Problem-Solving and Polynomial Equations-Solving 一书. 本书是围绕作者命名的数学机械化这一中心议题而陆续发表的一系列论文的综述. 本书试图以构造性与算法化的方式来研究数学, 使数学推理机械化以至于自动化, 由此减轻繁琐的脑力劳动.
全书分成三个部分：*部分考虑数学机械化的发展历史, 特别强调在古代中国的发展历史. 第二部分给出求解多项式方程组所依据的基本原理与特征列方法. 作为这一方法的基础, 本书还论述了构造性代数几何中的若干问题. 第三部分给出了特征列方法在几何定理证明与发现、机器人、天体力学、全局优化和计算机辅助设计等领域中的应用.
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Contents
Chapter 1 Polynomial Equations-Solving in Ancient Times, Mainly in Ancient China 1
1.1 A Brief Description of History of Ancient China and Mathematics Classics in Ancient China 1
1.2 Polynomial Equations-Solving in Ancient China 9
1.3 Polynomial Equations-Solving in Ancient Times beyond China and the Program of Descartes 24
Chapter 2 Historical Development of Geometry Theorem-Proving and Geometry Problem-Solving in Ancient Times 31
2.1 Geometry Theorem-Proving from Euclid to Hilbert 31
2.2 Geometry Theorem-Proving in the Computer Age 43
2.3 Geometry Problem-Solving and Geometry Theorem-Proving in Ancient China 47
Chapter 3 Algebraic Varieties as Zero-Sets and Characteristic-Set Method 65
3.1 Affine and Projective SpaceExtended Points and Specialization 65
3.2 Algebraic Varieties and Zero-Sets 73
3.3 Polsets and Ascending SetsPartial Ordering 85
3.4 Characteristic Set of a Polset and Well-Ordering Principle 93
3.5 Zero-Decomposition Theorems 104
3.6 Variety-Decomposition Theorems 117
Chapter 4 Some Topics in Computer Algebra 130
4.1 Tuples of integers 130
4.2 Well-Arranged Basis of a Polynomial Ideal 138
4.3 Well-Behaved Basis of a Polynomial Idea l45
4.4 Properties of Well-Behaved Basis and its Relationship with Groebner Basis 153
4.5 Factorization and GCD of Multivariate Polynomials over Arbitrary Extension Fields 164
Chapter 5 Some Topics in Computational Algebraic Geometry 175
5.1 Some Important Characters of Algebraic Varieties Complex and Real Varieties 175
5.2 Algebraic Correspondence and Chow Form 190
5.3 Chern Classes and Chern Numbers of an Irreducible Algebraic Variety with Arbitrary Singularities 202
5.4 A Projection Theorem on Quasi-Varieties 211
5.5 Extremal Properties of Real Polynomials 220
Chapter 6 Applications to Polynomial Equations-Solving 234
6.1 Basic Principles of Polynomial Equations-Solving: The Char-Set Method 234
6.2 A Hybrid Method of Polynomial Equations-Solving 244
6.3 Solving of Problems in Enumerative Geometry 256
6.4 Central Configurations in Planet Motions and Vortex Motions 266
6.5 Solving of Inverse Kinematic Equations in Robotics 277
Chapter 7 Appicaltions to Geometry Theorem-Proving 290
7.1 Basic Principles of Mechanical Geometry Theorem-Proving 290
7.2 Mechanical Proving of Geometry Theorems of Hilbertian Type 301
7.3 Mechanical Proving of Geometry Theorems involving Equalities Alone 316
7.4 Mechanical Proving of Geometry Theorems involving Inequalities 327
Chapter 8 Diverse Applications 341
8.1 Applications to Automated Discovering of Unknown Relations and Automated Determination of Geometric Loci 341
8.2 yApplications to Problems involving Inequalities, Optimization Problems, and Non-Linear Programming 353
8.3 Applications to 4-Bar Linkage Design 363
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