Polynomial algebra offers a standard, powerful, and robust approach to handle several important problems in geometric modelling and other areas. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the existence of multiple (or degenerate) roots. We describe discriminants in a general context, and relate them to an equally useful object, namely the resultant of an overconstrained polynomial system. We discuss several relevant applications in geometric modelling so as to motivate the use of such algebraic tools in further geometric problems. We focus on exploiting the sparseness of polynomials via the theory of Newton polytopes and sparse elimination.

Sparse Discriminants and Applications

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.

Determining a generalized Maxwell set

Given a polynomial in a single variable, we can find algebraic conditions on the coefficients corresponding to geometrically apparent features of its graph. For example, in elementary catastrophe theory, a bifurcation set in the coefficient space corresponds to a flat point of inflection on the curve. In this paper we establish algebraic conditions for the polynomial to have a pair of minima that have a common horizontal tangent. This condition is exactly that required by the Maxwell convention of catastrophe theory. The first step is a proof of a result giving a factorization of the discriminant of the discriminant of a polynomial. The proof given is a direct algebraic factorization. This result provides a condition for equal critical values. To narrow down to the cases in which the equal values are both minima, a method of using a combination of Sturm sequences has been devised. The paper thus provides complete rigorous algorithm for the determination of an algebraic expression for the Maxwell set of a ...

A.N. Godwin

Papers

The precise determination of Maxwell sets for cuspoid catastrophes