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1．Geometry,Algebra,and Algorithms 1

1．Polynomials and Affine Space 1

2．Affine Varieties 5

3．Parametrizations of Affine Varieties 14

4．Ideals 29

5．Polynomials of One Variable 38

2．Groebner Bases 49

1．Introduction 49

2．Orderings on the Monomials in k[x1,...,xn] 54

3．A Division Algorithm in k[x1...,xn] 61

4．Monomial Ideals and Dickson's Lemma 69

5．The Hilbert Basis Theorem and Groebner Bases 75

6．Properties of Groebner Bases 82

7．Buchberger's Algorithm 88

8．First Applications of Groebner Bases 95

9．(Optional)Improvements on Buchberger's Algorithm 102

3．Elimination Theory 115

1．The Elimination and Extension Theorems 115

2．The Geometry of Elimination 123

3．Implicifization 128

4．Singular Points and Envelopes 137

5．Unique Factorization and Resultants 150

6．Resultants and the Extension Theorem 162

4．The Algebra-Geometry Dictionary 169

1．Hilbert's Nullstellensatz 169

2．Radical Ideals and the Ideal-Variety Correspondence 175

3．Sums,Products,and Intersections of Ideals 183

4．Zariski Closure and Quotients of Ideals 193

5．Irreducible Varieties and Prime Ideals 198

6．Decomposition of a Variety into Irreducibles 204

7．(Optional)Primary Decomposition of Ideals 210

8．Summary 214

5．Polynomial and Rational Functions on a Variety 215

1．Polynomial Mappings 215

2．Quotients of Polynomial Rings 221

3．Algorithmic Computations in k[x1...,xn]/I 230

4．The Coordinate Ring of an Affine Variety 239

5．Rational Functions on a Variety 248

6．(Optional)Proof of the Closure Theorem 258

6．Robotics and Automatic Geometric Theorem Proving 265

1．Geometric Description of Robots 265

2．The Forward Kinematic Problem 271

3．The Inverse Kinematic Problem and Motion Planning 279

4．Automatic Geometric Theorem Proving 291

5．Wu's Method 307

7．Invariant Theory of Finite Groups 317

1．Symmetric Polynomials 317

2．Finite Matrix Groups and Rings of Invariants 327

3．Generators for the Ring of Invariants 336

4．Relations Among Generators and the Geometry of Orbits 345

8．Projective Algebraic Geometry 357

1．The Projective Plane 357

2．Projective Space and Projective Varieties 368

3．The Projective Algebra-Geometry Dictionary 379

4．The Projective Closure of an Affine Variety 386

5．Projective Elimination Theory 393

6．The Geometry of Quadric Hypersurfaces 408

7．Bezout's Theorem 422

9．The Dimension of a Variety 439

1．The Variety of a Monomial Ideal 439

2．The Complement of a Monomial Ideal 443

3．The Hilbert Function and the Dimension of a Variety 456

4．Elementary Properties of Dimension 468

5．Dimension and Algebraic Independence 477

6．Dimension and Nonsingularity 484

7．The Tangent Cone 495

Appendix A．Some Concepts from Algebra 509

1．Fields and Rings 509

2．Groups 510

3．Determinants 511

Appendix B．Pseudocode 513

1．Inputs,Outputs,Variables,and Constants 513

2．Assignment Statements 514

3．Looping Structures 514

4．Branching Structures 515

Appendix C．Computer Algebra Systems 517

1．AXIOM 517

2．Maple 520

3．Mathematica 522

4．REDUCE 524

5．Other Systems 528

Appendix D．Independent Projects 530

1．General Comments 530

2．Suggested Projects 530

References 535

Index 541