Implicitization of a General Union of Parametric Varieties

We reconduct the computation of the equations defining a union V of parametric varieties, up to a given degree d, to the computation of the equations, of degree ≤ d, vanishing on a finite set of points of V. If V is general this gives a new algorithm for implicitizingV. We have compared the implementation of this algorithm with the Hilbert-driven elimination algorithm included in the software packages CoCoA 3.7 and Singular 1.2, obtaining significant time savings. Moreover, the implementation of the algorithm leads to two intriguing conjectures on the natural number of generators of the union of general parametric curves or surfaces.