Discriminants, Resultants, and Multidimensional Determinants

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.

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).

In Chapter 1 we introduced, for any projective variety X ⊂ P 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).

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.

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.

In Part I we studied discriminants and resultants in the general context of projective geometry: our setup was that of an arbitrary projective variety X ⊂ P 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 X ⊂ P 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.

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.

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.

We now introduce the second main object of study: the A-discriminant Δ _A .

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:

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 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 .

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 }

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.

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].