Toric resultants and applications to geometric modelling

Summary

Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.

We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.

Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.