On the simplification of the coefficients of a parametrization

This paper deals with the problem of finding, for a given parametrization of an algebraic variety V of arbitrary dimension, another parametrization with coefficients over a smaller field. We proceed adapting, to the parametric case, a construction introduced by A. Weil for implicitly given varieties. We find that this process leads to the consideration of new varieties of a particular kind (ultraquadrics, in the terminology of this paper) in order to check, algorithmically, several interesting properties of the given variety V, such as the property of being reparametrizable over the smaller field.