Determining a generalized Maxwell set

Abstract

In elementary catastrophe theory certain polynomials are studied and algebraic conditions on the coefficients of these polynomials are derived. These conditions define curves, surfaces, etc. in the spaces of the coefficients. The condition that the polynomial has a degenerate critical point defines the standard bifurcation set and provides the well known cusp diagram in the coefficient space of the cusp catastrophe. In this paper we consider the condition that a polynomial has critical values that differ by a given amount. Using the discriminant of the discriminant of a polynomial and the algebraic theory of roots of a polynomial an algorithm is defined for deriving an algebraic expression corresponding to this condition. The consequences of the algorithm are given for the cusp catastrophe and diagrams in the coefficient space are drawn. The implications of the diagrams in terms of state transitions are derived, thus providing the starting point for a new range of applications of the cusp catastrophe.