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## Über dieses Buch

“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

## Inhaltsverzeichnis

### Introduction

Abstract
In this book we study discriminants and resultants of polynomials in several variables. The most familiar example is the discriminant of a quadratic polynomial f(x) = ax 2 + bx + c.
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 1. Projective Dual Varieties and General Discriminants

Abstract
We denote by P n the standard complex projective space of dimension n. Thus a point of P n is given by (n + 1) homogeneous coordinates (x 0:...: x n ), x i ∈ C, which are not all equal to 0 and are regarded modulo simultaneous multiplication by a non-zero number. More generally, if V is a finite-dimensional complex vector space, then we denote by P(V) the projectivization of V, i.e., the set of 1-dimensional vector subspaces in V. Thus P n = P(C n+1).
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 2. The Cayley Method for Studying Discriminants

Abstract
In Chapter 1 we introduced, for any projective variety XP n , the X-discriminant Δ X which is the equation of the projective dual variety X (so Δ X is a constant if X is not a hypersurface). We now explain the method that allows us, for a smooth X, to write down at least in principle, the polynomial Δ X . The method goes back to the remarkable paper by Cayley [Ca4] on elimination theory, in which the foundations were laid for what is now called homological algebra. The Cayley method can also be applied to other similar problems, such as finding resultants (see Chapter 3).
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 3. Associated Varieties and General Resultants

Abstract
The Grassmann variety (or Grassmannian) G(k, n) is the set of all k-dimensional vector subspaces in C n . For k = 1, this is the projective space P n−1. Since vector subspaces in C n correspond to projective subspaces in P n−1, we see that G(k, n) parametrizes (k−1)-dimensional projective subspaces in P n−1. In a more invariant fashion, we can start from any finite-dimensional vector space V and construct the Grassmannian G(k, V) of k-dimensional vector subspaces in V.
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 4. Chow Varieties

Abstract
The Grassmann variety G(k, h) parametrizes (k −1)-dimensional projective subspaces in P n−1. Projective subspaces are just algebraic subvarieties of degree 1. It is natural to look for parameter spaces parametrizing subvarieties of a given degree d ≥ 1. Here, however, we encounter some new phenomena. Namely, an irreducible variety can degenerate into a reducible one (e.g., a curve can degenerate into a collection of straight lines). Moreover, consider a reducible variety, say, a union of two distinct lines. Such a variety can degenerate into one line, which apparently has a smaller degree. Of course, in this case it is natural to count the limiting line with multiplicity 2. To take into account all of these possibilities, we need the notion of an algebraic cycle.
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 5. Toric Varieties

Abstract
In Part I we studied discriminants and resultants in the general context of projective geometry: our setup was that of an arbitrary projective variety XP n−1. We now want to move into a more combinatorial setting, which is closer to the classical concept of discriminants and resultants for polynomials. This setting corresponds to the situation when XP n−1 is a toric variety. In the present chapter, we have adapted the theory of toric varieties for our purposes. Since there are several references available on the subject [D] [Fu 2] [O], we did not attempt to be exhaustive or self-contained. Our exposition is organized “from the special to the general” so that the general description of toric varieties in terms of fans appears at the very end of the chapter.
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 6. Newton Polytopes and Chow Polytopes

Abstract
Suppose we have a complicated (Laurent) polynomial f(x 1,..., x k ) in k variables. Let A be the set of monomials in f with non-zero coefficients. As we have seen in Chapter 5, to understand the structure of f, it is natural to consider it as a member of the space C A of all polynomials whose monomials belong to A.
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 7. Triangulations and Secondary Polytopes

Abstract
In this chapter we discuss a combinatorial framework for discriminants and resultants related to toric varieties. The main construction introduces a certain class of polytopes, called secondary polytopes, whose vertices correspond to certain triangulations of a given convex polytope. These polytopes will play a crucial role later in the study of the Newton polytopes of discriminants and resultants. The constructions in this chapter are quite elementary.
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 8. A-Resultants and Chow Polytopes of Toric Varieties

Abstract
We begin, starting with resultants, to apply the general formalism of Part I to discriminants and resultants associated with toric varieties. The treatment of discriminants is left for the next chapter.
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 9. A-Discriminants

Abstract
We now introduce the second main object of study: the A-discriminant Δ A .
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 10. Principal A-Determinants

Abstract
Our aim in this and in the following chapter is to study the Newton polytope of the A-discriminant Δ A . This will be done through an intermediary object, the so-called principal A-determinant E A . Like the A-discriminant, E A = E A (f) is a polynomial function in coefficients a ω of an indeterminate polynomial f ∈ C A . We shall do the following:
(1)
give a complete description of the Newton polytope of E A . It turns out to coincide with the secondary polytope Σ(A) (see Chapter 7);

(2)
give a formula (prime factorization) expressing E A as a product of Δ A and discriminants corresponding to some subsets of A;

(3)
give a formula for the product of values of a polynomial at its critical points in terms of the principal A-determinants.

Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 11. Regular A-Determinants and A-Discriminants

Abstract
In the previous chapter we established some structural properties of the principal A-determinant E A . Now we shall apply this information to the A-discriminant Δ A . In the most important case when the toric variety X A is smooth, we have
$${E_A}(f) = \prod\limits_{\Gamma \subset Q} {{\Delta _{A \cap \Gamma }}} (f)$$
where the product is taken over all the faces of the polytope Q = Conv (A) (Theorem 1.2 Chapter 10). Since (in the case when X A is smooth) a similar equality holds for each E A⋂Г, we have a system of equalities relating the polynomials Δ A⋂Г and E A⋂Г that allows us to recover Δ A as as an alternating product of the E A⋂Г. Consequently, alternating sums and products will appear in the expressions for the Newton polytope and coefficients of Δ A .
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 12. Discriminants and Resultants for Polynomials in One Variable

Abstract
In this chapter we consider the most classical case of the discriminant of a polynomial in one variable, and the resultant of two such polynomials. In the general language of Part II, we consider the A-discriminant Δ A where A consists of monomials 1, x, x 2,..., x n and the (A 1, A 2)-resultant where A 1 = {1, x,..., x m } and A 2 = {1, x,..., x A }
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 13. Discriminants and Resultants for Forms in Several Variables

Abstract
We now consider the most straightforward generalization of the resultants and discriminants treated in Chapter 12, namely the resultants and discriminants for homogeneous forms in several variables. Although they are very classical objects of study, many fundamental questions about them still remain open. To realize how little is known it is enough to mention that an explicit polynomial expression still remains unknown with the exception of some very special cases. The methods developed in Parts I and II help to put this and other questions in a general perspective. Without pretending to be complete, we present an overview of some fairly classical results together with more fresh developments.
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Chapter 14. Hyperdeterminants

Abstract
The goal of this chapter is to provide a natural “higher dimensional” generalization of the classical notion of the determinant of a square matrix. There were some attempts toward a rather straightforward definition of the “hyperdeterminant” for “hypercubic” matrices using alternating summations over the product of several symmetric groups (see e.g., [P], §54 and references therein). Here we systematically develop another approach under which the hyperdeterminant becomes a special case of the general discriminant studied in the previous chapters. As so many other ideas in the field, this approach is due to Cayley [Ca1].
Israel M. Gelfand, Mikhail M. Kapranov, Andrei V. Zelevinsky

### Backmatter

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