Discriminants, Resultants, and Multidimensional Determinants (Google eBook)

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Springer, Apr 16, 2008 - Mathematics - 523 pages
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"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
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