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. © 2011 Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg.