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This paper presents a new algorithm for implicitizing tensor product surfaces of bi-degree ( m , n ) with no base points, assuming that there are no moving planes of bi-degree ( m − 1 , n − 1 ) following the surface. The algorithm is based on some structural results: (1) There are exactly 2 n linearly independent moving planes of bi-degree ( m , n − 1 ) following the surface; (2) mn linearly independent moving quadrics of bi-degree ( m − 1 , n − 1 ) following the surface can be constructed from the 2 n linearly independent moving planes; (3) The mn linearly independent moving quadrics form a compact determinant of order mn which exactly gives the implicit equation of the rational surface. Complexity analysis and experimental results show that the new algorithm is significantly more efficient than the previous methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose algorithms for the following problems in the implicitization of a set of differential rational parametric equations P. (1) To find a characteristic set for the implicit prime ideal of P. (2) To find a canonical representation for the image of P. (3) To decide whether the parameters of P are independent, and if not, to re-parameterize P so that the new parametric equations have independent parameters. (4) To compute the inversion maps of P, and as a consequence, to decide whether P is proper. If the implicit variety is of dimension one and P is not proper, to find a proper re-parameterization for P. [Copyright &y& Elsevier]