Equations of parametric surfaces with base points via syzygies

Let S be a tensor product parametrized surface in P-3; that is, S is given as the image of phi : P-1 x P-1 --> P-3. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of S when certain base points are present. This work extends the algorithm provided by Cox [Cox, D.A., 2001. Equations of parametric curves and surfaces via syzygies. In: Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering. Contemporary Mathematics vol. 286, pp. 1-20] for when phi has no base points, and it is analogous to some of the results of Buse et al. [Buse, L., Cox, D., D' Andrea, C., 2003. Implicitization of surfaces in P-3 in the presence of base points. J. Algebra Appl. 2 (2), 189-214] for the case of a triangular parametrization phi : P-2 --> P-3 with base points. (C) 2004 Elsevier Ltd. All rights reserved.