●Chapter 1 Polynomial Equations-Solving in Ancient Times， Mainly in Ancient China
1.1 A Brief Description of History of Ancient China and Mathematics Classics in Ancient China
1.2 Polynomial Equations-Solving in Ancient China
1.3 Polynomial Equations-Solving in Ancient Times beyond China and the Program of Descartes
Chapter 2 Historical Development of Geometry Theorem-Proving and Geometry Problem-Solving in Ancient Times
2.1 Geometry Theorem-Proving from Euclid to Hilbert
2.2 Geometry Theorem-Proving in the Computer Age
2.3 Geometry Problem-Solving and Geometry Theorem-Proving in Ancient China
Chapter 3 Algebraic Varieties as Zero-Sets and Characteristic-Set Method
3.1 Affine and Projective SpaceExtended Points and Speization
3.2 Algebraic Varieties and Zero-Sets
3.3 Polsets and Ascending SetsPartial Ordering
3.4 Characteristic Set of a Polset and Well-Ordering Principle
3.5 Zero-Decomposition Theorems
3.6 Variety-Decomposition Theorems
Chapter 4 Some Topics in Computer Algebra
4.1 Tuples of integers
4.2 Well-Arranged Basis of a Polynomial Ideal
4.3 Well-Behaved Basis of a Polynomial Idea l
4.4 Properties of Well-Behaved Basis and its Relationship with Groebner Basis
4.5 Factorization and GCD of Multivariate Polynomials over Arbitrary Extension Fields
Chapter 5 Some Topics in Computational Algebraic Geometry
5.1 Some Important Characters of Algebraic Varieties Complex and Real Varieties
5.2 Algebraic Correspondence and Chow Form
5.3 Chern Classes and Chern Numbers of an Irreducible Algebraic Variety with Arbitrary Singularities
5.4 A Projection Theorem on Quasi-Varieties
5.5 Extremal Properties of Real Polynomials
Chapter 6 Applications to Polynomial Equations-Solving
6.1 Basic Principles of Polynomial Equations-Solving: The Char-Set Method
6.2 A Hybrid Method of Polynomial Equations-Solving
6.3 Solving of Problems in Enumerative Geometry
6.4 Central Configurations in Planet Motions and Vortex Motions
6.5 Solving of Inverse Kinematic Equations in Robotics
Chapter 7 Appicaltions to Geometry Theorem-Proving
7.1 Basic Principles of Mechanical Geometry Theorem-Proving
7.2 Mechanical Proving of Geometry Theorems of Hilbertian Type
7.3 Mechanical Proving of Geometry Theorems involving Equalities Alone
7.4 Mechanical Proving of Geometry Theorems involving Inequalities
Chapter 8 Diverse Applications
8.1 Applications to Automated Discovering of Unknown Relations and Automated Determination of Geometric Loci
8.2 yApplications to Problems involving Inequalities， Optimization Problems， and Non-Linear Programming
8.3 Applications to 4-Bar Linkage Design
8.4 Applications to Surface-Fitting Problem in CAGD
8.5 Some Miscellaneous Complements and Extensions
Bibliography
Index
●
●
內容簡介
本卷收錄了吳文俊的Mathematics Mechanization:Mechanical Geometry Theorem-Proving，Mechanical Geometry Problem-Solving and Polynomial Equations-Solving一書。本書是圍繞作者命名的“數學機械化”這一中心議題而陸續發表的一繫列論文的綜述。本書試圖以構造性與算法化的方式來研究數學，使數學推理機械化以至於自動化，由此減輕繁瑣的腦力勞動。全書分成三個部分：部分考慮數學機械化的發展歷史，特別強調在古代中國的發展歷史。第二部分給出求解多項式方程組所依據的基本原理與特征列方法。作為這一方法的基礎，本書還論述了構造性代數幾何中的若干問題。第三部分給出了特征列方法在幾何定理證明與發現、機器人、天體力學、全局優化和計算機輔助設計等領域中的應用。
