  According to several classical results by Bezout, Sylvester, Cayley, and
others, the classical discriminant D_n 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 D_n
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 x 1 matrix (D_n).
  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.
