Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Abstract

We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial.

Matrix Representations by Means of Interpolation

“…This idea has been extensively used [CGKW00,Dok01,MM02,BI89]. In [EKKL13,EKK15], sparse elimination theory is employed to predict the implicit monomials and build the interpolation matrix.…”

“…The method has been developed for plane curves, and (hyper)surfaces. The columns of the matrix are indexed by a superset of the implicit monomials, i.e., those in the implicit polynomial with non-zero coefficient [EKKL13]. This monomial set is determined quite tightly for parametric models, by means of the sparse resultant of the parametric polynomials, thus exploiting the sparseness of the parametric and implicit polynomials.…”

Minkowski Decomposition and Geometric Predicates in Sparse Implicitization

“…In [5] they show that if the kernel is not 1-dimensional then the predicted polytope is the Minkowski sum of the implicit polytope and an extraneous one. The true implicit polynomial is obtained as the greatest common divisor (GCD) of the polynomials corresponding to at least two and at most all of the kernel vectors, or via multivariate polynomial factoring.…”

“…Hence: Lemma 2. [5] Any polynomial in the basis of monomials S indexing M , with coefficient vector in the kernel of M , is a multiple of the implicit polynomial p(x).…”

“…In fact, it also accounts for them since then P is expected to be much smaller than Q. Theorem 3. [5] Let P = N (p(x)) be the implicit polytope, and Q be the predicted polytope. Assuming M has been built using sufficiently generic evaluation points, the dimension of its kernel equals r = #{a ∈ Z n+1 : a + P ⊆ Q} = #{a ∈ Z n+1 : N (x a · p(x)) ⊆ Q}.…”