A new determinantal formula for the classical discriminant

TL;DR

This paper introduces a novel determinantal formula for the classical discriminant of degree n polynomials, expanding beyond traditional equivalent formulas by constructing an explicit non-classical matrix representation.

Contribution

It presents an explicit non-classical determinantal formula for the discriminant, contrasting its properties with classical formulas and exploring its unique features.

Findings

Constructed a new determinantal formula using the open swallowtail matrix

Demonstrated differences between the new formula and classical ones

Analyzed properties and implications of the non-classical formula

Abstract

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.