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