No abstract

In this paper, we propose a new method, based on Bezoutian matrices, for computing a nontrivial multiple of the resultant over a projective variety X, which is described on an open subset by a parameterization. This construction, which generalizes the classical and toric one, also applies for instance to blowing up varieties and to residual intersection problems. We recall the classical notion of resultant over a variety X. Then we extend it to varieties which are parameterized on a dense open subset and give new conditions for the existence of the resultant over these varieties. We prove that any maximal nonzero minor of the corresponding Bezoutian matrix yields a nontrivial multiple of the resultant. We end with some experiments.

We give an overview of resultant theory and some of its applications in computer aided geometric design. First, we mention different formulations of resultants, including the projective resultant, the toric resultant, and the residual resultants. In the second part we illustrate these tools, and others projection operators, on typical problems as surface implicitization, inversion, intersection, and detection of singularities of a parameterized surface.ki j=0 c i,j ψ i,j (x) = 0 for i = 0, . . . , n. In other words, c is the first projection of the point (c, x) in the incidence variety W X = {(c, x) ∈ P k0 × · · · × P kn × X : f i (x) = 0, i = 0, . . . , n}.

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International audienceWe study rational ruled surfaces and $\mu$-bases which were recently considered in a series of articles by Chen and coworkers. We give short and conceptual proofs with geometric insights and efficient algorithms. In particular, we provide a method to reparameterize an improper parameterization and we also briefly explain how to deal with approximate input data. Finally we provide an algorithmic description of self-intersection loci