Implicitization Of Surfaces

There are following problems in the implicization of surfaces. (1) Eliminate the prameters of a set of rational parametic equations;(2)Find a minimumal variety that contains rational parametric surfaces.In this thesis, we use Groebner bases theory to study the implicization of surfaces, In fact,the parameter equations have defined a map F:P^m(k)→Pⁿ(k), the implicit equation is a minimumal variety that contains rational parametric surfaces. If k is an infinite field,We have proved: (1) If base points B =Φ, then F(P^m) = V(I_m+1) ;(2) If base points B≠Φ, thenF(P^m\B)(?)V(I_m+1), and V(I_m+1) is a minimumal variety that contains F(P^m \ B).