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Discriminants, resultants, and multidimensional determinants - Gelfand I. M., Kapranov M. M., Zelevinsky A. V.

 
Название: Discriminants, resultants, and multidimensional determinants
Автор: Gelfand I. M., Kapranov M. M., Zelevinsky A. V.
Категория: Математика
Тип: Книга
Дата: 30.12.2008 19:09:13
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Описание: This book has expanded from our attempt to construct a general theory of hyper-geometric functions and can be regarded as a first step towards its systematic exposition. However, this step turned out to be so interesting and important, and the whole program so overwhelming, that we decided to present it as a separate work. Moreover, in the process of writing we discovered a beautiful area which had been nearly forgotten so that our work can be regarded as a natural continuation of the classical developments in algebra during the 19th century. We found that Cayley and other mathematicians of the period understood many of the concepts which today are commonly thought of as modern and quite recent. Thus, in an 1848 note on the resultant, Cayley in fact laid out the foundations of modern homological algebra. We were happy to enter into spiritual contact with this great mathematician. The place of discriminants in the general theory of hypergeometric functions is similar to the place of quasi-classical approximation in quantum mechanics. More precisely, in [GGZ] [GKZ2] [GZK1 ] a general class of special functions was introduced and studied, the so-called /l-hypergeometric functions. These functions satisfy a certain holonomic system of linear partial differential equations (the Л-hypergeometric equations). The A -discriminant, which is one of our main objects of study, describes singularities of Л-hypergeometric functions. According to the general principles of the theory of linear differential equations, these singularities are governed by the vanishing of the highest symbols of A-hypergeometric equations. The relation between differential operators and their highest symbols is the mathematical counterpart of the relation between quantum and classical mechanics; so we can say that hypergeometric functions provide a "quantization" of discriminants. In our work on hypergeometric functions we found connections with many questions in algebra and combinatorics. We hope that this book brings to light some of these connections. One of the algebraic concepts which seems to us particularly important is that of hyperdeterminants (analogs of determinants for multi-dimensional "matrices.") After rediscovering hyperdeterminants in connection with hypergeometric functions, we found that they too, had been introduced by Cayley in the 1840s. Unfortunately, later on, the study of hyperdeterminants was largely abandoned in favor of another, more straightforward definition (cf. [P]). The only other work on hyperdeterminants of which we are aware is an important
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