Equations of parametric surfaces with base points via syzygies

Authors:
William A. Adkins;J. William Hoffman;Hao Hao Wang
Affiliations:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, United States;Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, United States;Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO 63701, United States
Venue:
Journal of Symbolic Computation
Year:
2005

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Abstract

Let S be a tensor product parametrized surface in P^3; that is, S is given as the image of @f:P^1xP^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 @f 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 @f:P^2-P^3 with base points.