Proper reparametrization for inherently improper unirational varieties

Abstract

In this paper, a class of lattice supports in the lattice space Z m is found to be inherently improper because any rational parametrization from C m to C n defined on such a support is improper. The improper index for such a lattice support is defined to be the gcd of the normalized volumes of all the simplex sub-supports. The structure of an improper support S is analyzed and shrinking transformations are constructed to transform S to a proper one. For a generic rational parametrization RP defined on an improper support S, we prove that its improper index is the improper index of S and give a proper reparametrization algorithm for RP . Finally, properties for rational parametrizations defined on an improper support and with numerical coefficients are also considered.

Sparse difference resultant

“…Proof: This is a direct consequence of the Smith normal form method [4, p. 67]. Also see paper [32] for an alternative proof.…”

Numerical Proper Reparametrization of Space Curves and Surfaces

“…|S| ≥ 2, and the rank of M S is two, since the surface reduces to a curve if the rank is one. Furthermore, we assume that (0, 0) ∈ S without loss of generality (see Lemma 2 in [25]).…”

Numerical Reparametrization of Rational Parametric Plane Curves

“…Also, they provide an implicitization approach based on resultants (see [9] and [25]). Hence, the study of proper reparametrization has been concerned by some authors such as [5,17,18,25,27], and several efficient proper reparametrization algorithms can be found in [12,17,24].…”