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Introduction 1

Ⅰ．GENERAL DISCRIMINANTS AND RESULTANTS 13

CHAPTER 1．Projective Dual Varieties and General Discriminants 13

1．Definitions and basic examples 13

2．Duality for plane curves 16

3．The incidence variety and the proof of the biduality theorem 27

4．Further examples and properties of projective duality 30

5．The Katz dimension formula and its applications 39

CHAPTER 2．The Cayley Method for Studying Discriminants 48

1．Jet bundles and Koszul complexes 48

2．Discriminantal complexes 54

3．The degree and the dimension of the dual 61

4．Discriminantal complexes in terms of differential forms 71

5．The discriminant as the determinant of a spectral sequence 80

CHAPTER 3．Associated Varieties and General Resultants 91

1．Grassmannians.Preliminary material 91

2．Associated hypersurfaces 97

3．Mixed resultants 105

4．The Cayley method for the study of resultants 112

CHAPTER 4．Chow Varieties 122

1．Definitions and main properties 122

2．O-cycles,factorizable forms and symmetric products 131

3．Cayley-Green-Morrison equations of Chow varieties 146

Ⅱ．A-DISCRIMINANTS AND A-RESULTANTS 165

CHAPTER 5．Toric Varieties 165

1．Projectively embedded toric varieties 165

2．Affine toric varieties and semigroups 172

3．Local structure of toric varieties 177

4．Abstract toric varieties and fans 187

CHAPTER 6．Newton Polytopes and Chow Polytopes 193

1．Polynomials and their Newton polytopes 193

2．Theorems of Kouchnirenko and Bernstein on the number of solutions of a system of equations 200

3．Chow polytopes 206

CHAPTER 7．Triangulations and Secondary Polytopes 214

1．Triangulations and secondary polytopes 214

2．Faces of the secondary polytope 227

3．Examples of secondary polytopes 233

CHAPTER 8．A-Resultants and Chow Polytopes of Toric Varieties 252

1．Mixed(Al,...,Ak)-resultants 252

2．The A-resultant 255

3．The Chow polytope of a toric variety and the secondary polytope 259

CHAPTER 9．A-Discriminants 271

1．Basic definitions and examples 271

2．The discriminantal complex 275

3．A differential-geometric characterization of A-discriminantal hypersurfaces 285

CHAPTER 10．Principal A-Determinants 297

1．Statements of main results 297

2．Proof of the prime factorization theorem 313

3．Proof of the properties of generalized A-determinants 329

4．The proof of the product formula 333

CHAPTER 11．Regular A-Determinants and A-Discriminants 344

1．Differential forms on a singular toric variety and the regular A-determinant 344

2．Newton numbers and Newton functions 351

3．The Newton polytope of the regular A-determinant and D-equivalence of triangulations 361

4．More on D-equivalence 370

5．Relations to real algebraic geometry 378

Ⅲ．CLASSICAL DISCRIMINANTS AND RESULTANTS 397

CHAPTER 12．Discriminants and Resultants for Polynomials in One Variable 397

1．An overview of classical formulas and properties 397

2．Newton polytopes of the classical discriminant and resultant 411

CHAPTER 13．Discriminants and Resultants for Forms in Several Variables 426

1．Homogeneous forms in several variables 426

2．Forms in several groups of variables 437

CHAPTER 14．Hyperdeterminants 444

1．Basic properties of the hyperdeterminant 444

2．The Cayley method and the degree 450

3．Hyperdeterminant of the boundary format 458

4．Schl？fli's method 475

APPENDIX A．Determinants of Complexes 480

APPENDIX B．A.Cayley:On the Theory of Elimination 498

Bibliography 503

Notes and References 513

List of Notations 517

Index 521