A New Determinantal Formula for the Classical Discriminant

According to several classical results by Bézout, Sylvester, Cayley, and others, the classical discriminant Dn of degree n polynomials may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f . However, all of the determinantal formulae for Dn appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 × 1 matrix (Dn). In this paper, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol’d and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae. Let f(x, y) := a0x + a1xy + · · · + an−1xy + any be a homogeneous binary form of degree n over an algebraically closed field K of characteristic zero. Denote by V the K-vector space of dimension n + 1 with basis {a0, . . . , an}. We may identify forms f , up to nonzero scalar multiple, with points in the projective space P(V ). For α, β ∈ K, not both zero, we say that the point [α : β] ∈ PK is a root of f of multiplicity k if the polynomial (βx− αy) divides f . Definition 1. The (projective) classical discriminant of degree n polynomials, denoted ∆n, or ∆ when n is understood, is the locus of forms f with a root of multiplicity at least two. The variety ∆ is a hypersurface in P(V ) whose defining squarefree polynomial in K[a0, . . . , an] we call Dn. There is much interest in evaluating Dn on a given binary form or on a family thereof. The direct approach — constructing Dn via, say, a parametrization of ∆ and writing Dn explicitly — is infeasible for large values of n, since the number of terms of Dn grows very quickly. With this in mind, we are interested in the construction of a determinantal formula for Dn, that is, a square matrix over K[a0, . . . , an] whose determinant is Dn. Of course, there exists a trivial formula, namely, the 1× 1 matrix whose entry is Dn. More generally, we shall refer to any formula with matrix A as trivial if either A or its classical adjoint is invertible. There exist several nontrivial classical such formulae by such mathematicians as Bézout, Sylvester, and Cayley. However, they are all equivalent in the sense that 1With respect to the Z3 grading deg ai := (1, n−i, i), Dn is homogeneous of degree (2(n−1), n(n− 1), n(n − 1)). Thus one can compute an upper bound for the number of terms in Dn which is exponential in n. There is no reason to believe that this bound is not tight, since it is unlikely that many of the coefficients of the terms of degree (2(n− 1), n(n− 1), n(n− 1)) vanish. 1 ar X iv :0 81 2. 11 97 v2 [ m at h. A G ] 2 7 Ja n 20 09