Compact Formulae in Sparse Elimination

International audienceIt has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bézout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects

Geometric operations using sparse interpolation matrices

Based on the computation of a superset of the implicit support, implicitization of a parametrically given hypersurface is reduced to computing the nullspace of a numeric matrix. Our approach predicts the Newton polytope of the implicit equation by exploiting the sparseness of the given parametric equations and of the implicit polynomial, without being affected by the presence of any base points. In this work, we study how this interpolation matrix expresses the implicit equation as a matrix determinant, which is useful for certain operations such as ray shooting, and how it can be used to reduce some key geometric predicates on the hypersurface, namely membership and sidedness for given query points, to simple numerical operations on the matrix, without need to develop the implicit equation. We illustrate our results with examples based on our Maple implementation. © 2015 Elsevier Inc. All rights reserved