In many applications, where geometric constructions appear, parametrizations of the geometric objects are used. In practice, when the geometric entities are algebraic, the varieties are assumed to be rational, and hence representable by means of rational functions. This is a limitation of the feasibility of the application, since the class of
rational varieties may turn to be small for the practical purposes. To overcome this difficulty one may use approximations or, alternatively, enlarge the family of functions allowed to be used in the parametric representation of the geometric object, for instance by introducing radicals of polynomials. This is the theoretical frame of this talk.
In this talk, we will introduce the notion of rational parametrization as well as the associated concept of rational variety. Additionally, we will introduce a second, auxiliary, variety, named the tower variety. In addition, we will study some of their fundamental properties. From the computational point of view, we will present algorithms for implicitizing radical varieties and, for the particular cases of curves and
surfaces, we will also present reparametrization and parametrization algorithms.