A first approach towards normal parametrizations of algebraic surfaces

Abstract

In this paper we analyze the problem of deciding the normality (i.e. the surjectivity) of a rational parametrization of a surface S. The problem can be approached by means of elimination theory techniques, providing a proper close subset Z ⊂ S where surjectivity needs to be analyzed. In general, these direct approaches are unfeasible because Z is very complicated and its elements computationally hard to manipulate. Motivated by this fact, we study ad hoc computational alternative methods that simplifies Z. For this goal, we introduce the notion of pseudo-normality, a concept that provides necessary conditions for a parametrization for being normal. Also, we provide an algorithm for deciding the pseudo-normality. Finally, we state necessary and sufficient conditions on a pseudo-normal parametrization to be normal. As a consequence, certain types of parametrizations are shown to be always normal. For instance, pseudo-normal polynomial parametrizations are normal. Moreover, for certain class of parametrizations, we derive an algorithm for deciding the normality.