Discriminants, Resultants, and Multidimensional Determinants

Inhaltsverzeichnis

Preface.- Introduction.- General Discriminants and Resultants.- Projective Dual Varieties and General Discriminants.- The Cayley Method of Studying Discriminants.- Associated Varieties and General Resultants.- Chow Varieties.- Toric Varieties.- Newton Polytopes and Chow Polytopes.- Triangulations and Secondary Polytopes.- A-Resultants and Chow Polytopes of Toric Varieties.- A-Discriminants.- Principal A-Discriminants.- Regular A-Determinants and A-Discriminants.- Classical Discriminants and Resultants.- Discriminants and Resultants for Polynomials in One Variable.- Discriminants and Resultants for Forms in Several Variables.- Hyperdeterminants.- Appendix A. Determinants.- Appendix B. A. Cayley: On the Theory of Elimination.- Bibliography.- Notes and References.- List of Notation.- Index.

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From the reviews:

"The book has developed into an indispensable modern classic, during the past 15 years, and it has initiated a true avalanche of related research activities due to its pioneering and inspiring impact. The steadily large number of quotations, which seems to be ever-flowing, bespeaks the undiminished pacemaking role of this unrivalled source book in various areas of current mathematical research, ranging from algebraic geometry and the general theory of hypergeometric functions up to their recent applications in mathematical physics...

It is and remains the broader, unifying approach combining the classical heritage and the powerful abstract viewpoint of the subject that makes the book under review so timeless, attractive, enlightening, inspiring and virtually invaluable -- a characteristic that will persist for further decades in the future. Besides, the comparatively inexpensive reprint of this classic in paperback form must be seen as another rewarding service to the mathematical community as a whole." -Zentralblatt Math (Review of the Softcover Edition)

"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 (Review of the Hardcover Edition)

"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 (Review of the Hardcover Edition)

"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 (Review of the Hardcover Edition)

"...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 (Review of the Hardcover Edition)

"It is very much representative of the Gelfand school style. ... Discriminants, Resultants, and Multidimensional Determinants is currently the most cited on MathSciNet. A perusal of the long list of citations indicates the enormous influence of the book. Many of the citations are clearly in the framework set up by the book." (David P. Roberts, The Mathematical Association of America, October, 2009)