Equations of parametric surfaces with base points via syzygies

Let S be a tensor product parametrized surface in P3; that is, S is given as the image of φ:P1×P1→P3. 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 φ has no base points, and it is analogous to some of the results of Busé et al. [Busé, L., Cox, D., D'Andrea, C., 2003. Implicitization of surfaces in P3 in the presence of base points. J. Algebra Appl. 2 (2), 189-214] for the case of a triangular parametrization φ:P2→P3 with base points.