Inversion, Degree and Reparametrization for Rational Surfaces

Abstract

Let ${\mathcal P}$ be a rational affine parametrization of an algebraic surface $\mathcal V$, and let $\phi_{\mathcal P}: {\rm IK}^{2} \longrightarrow {\mathcal V}; {\overline t} \longmapsto {\mathcal P}({\overline t})$ be the rational map induced by ${\mathcal P}$. In this survey, we consider three different problems. First we deal with the problem of deciding whether $\phi_{\mathcal P}$ is birational (i.e. whether ${\mathcal P}$ is proper); in case of birationality, the question of computing the inverse of the parametrization is considered. On the other side, the birationality of $\phi_{\mathcal P}$ is also characterized by . Hence the problem of analyzing the birationality is equivalent to computing . The second problem considered deals with this question. More precisely, we show that can be computed by means of greatest common divisor (gcd) and univariate resultant computations. Finally, if the given parametrization ${\mathcal P}$ is not proper and satisfies an additional condition, we solve the problem of proper reparametrization. That is, we determine a proper rational parametrization ${\mathcal Q}({\overline t})$ of $\mathcal V$ from ${\mathcal P}$ such that ${\mathcal P}({\overline t})={\mathcal Q}(R({\overline t}))$. All the results in this survey are included in Perez-Diaz et al. (2002), Perez-Diaz and Sendra (2004) or Perez-Diaz (2006).