A partial solution to the problem of proper reparametrization for rational surfaces

Given an algebraically closed field KK, and a rational parametrization PP of an algebraic surface V⊂K3V⊂K3, we consider the problem of computing a proper rational parametrization QQ from PP (reparametrization problem  ). More precisely, we present an algorithm that computes a rational parametrization QQ of VV such that the degree of the rational map induced by QQ is less than the degree induced by PP. The properness of the output parametrization QQ is analyzed. In particular, if the degree of the map induced by QQ is one, then QQ is proper and the reparametrization problem is solved. The algorithm works if at least one of two auxiliary parametrizations defined from PP is not proper.