## Discriminants, Resultants, and Multidimensional Determinants"This book revives and vastly expands the classical theory of resultants and discriminants. Most of the main new results of the book have been published earlier in more than a dozen joint papers of the authors. The book nicely complements these original papers with many examples illustrating both old and new results of the theory."—Mathematical Reviews "Collecting and extending the fundamental and highly original results of the authors, it presents a unique blend of classical mathematics and very recent developments in algebraic geometry, homological algebra, and combinatorial theory." —Zentralblatt Math "This book is highly recommended if you want to get into the thick of contemporary algebra, or if you wish to find some interesting problem to work on, whose solution will benefit mankind." —Gian-Carlo Rota, Advanced Book Reviews "...the book is almost perfectly written, and thus I warmly recommend it not only to scholars but especially to students. The latter do need a text with broader views, which shows that mathematics is not just a sequence of apparently unrelated expositions of new theories, ... but instead a very huge and intricate building whose edification may sometimes experience difficulties ... but eventually progresses steadily." —Bulletin of the American Mathematical Society |

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### Contents

ADISCRIMINANTS AND ARESULTANTS | 7 |

Projective Dual Varieties and General Discriminants | 12 |

The incidence variety and the proof of the biduality theorem | 27 |

The Cayley Method for Studying Discriminants | 48 |

The degree and the dimension of the dual | 61 |

The discriminant as the determinant of a spectral sequence I | 80 |

Associated hypersurfaces | 97 |

The Cayley method for the study of resultants | 112 |

AResultants and Chow Polytopes of Toric Varieties | 252 |

The Chow polytope of a toric variety and the secondary polytope | 259 |

ADiscriminants | 271 |

A differentialgeometric characterization | 285 |

Principal ADeterminants | 297 |

Proof of the prime factorization theorem | 313 |

Proof of the properties of generalized Adeterminants | 329 |

Regular ADeterminants and ADiscriminants | 344 |

Ocycles factorizable forms and symmetric products | 131 |

CayleyGreenMorrison equations of Chow varieties | 146 |

Toric Varieties | 163 |

Afﬁne toric varieties and semigroups | 172 |

Abstract toric varieties and fans | 187 |

Newton Polytopes and Chow Polytopes | 193 |

Theorems of Kouchnirenko and Bernstein on the number | 200 |

Chow polytopes | 206 |

Triangulations and Secondary Polytopes | 214 |

Faces of the secondary polytope | 227 |

Examples of secondary polytopes | 233 |

The Newton polytope of the regular Adeterminant | 361 |

Relations to real algebraic geometry | 378 |

Discriminants and Resultants for Polynomials in One Variable | 395 |

Newton polytopes of the classical discriminant and resultant | 411 |

Discriminants and Resultants for Forms in Several Variables | 426 |

Hyperdeterminants | 444 |

Hyperdeterminant of the boundary format | 458 |

Schliitlis method | 475 |

On the Theory of Elimination | 498 |

Notes and References | 513 |