Sparse Implicitization via Interpolation

Abstract

The aim of our work is to explore recent methods for computing the Newton polytope of the implicit equation and their applicability to implicitization by interpolation. We implement interpolation by exact or numerical linear algebra following an exact phase which computes a superset of the monomials appearing in the implicit equation. These monomials are then suitably evaluated to build a numeric matrix, ideally of corank 1, whose kernel vector contains the coefficients of the implicit equation. We propose techniques for handling the case of higher corank. This yields an efficient, output-sensitive algorithm for computing the implicit equation. The method can be applied to polynomial or rational parameterizations of planar curves or (hyper)surfaces of any dimension including parameterizations with base points. We conclude with certain extensions.