In this paper, we investigate some topics around the closed image S of a rational map λ given by some homogeneous elements f 1 , . . . , f n of the same degree in a graded algebra A. We first compute the degree of this closed image in case λ is generically finite and f 1 , . . . , f n define isolated base points in Proj(A). We then relate the definition ideal of S to the symmetric and the Rees algebras of the ideal I = (f 1 , . . . , f n ) ⊂ A, and prove some new acyclicity criteria for the associated approximation complexes. Finally, we use these results to obtain the implicit equation of S in case S is a hypersurface, Proj(A) = P n−2 k with k a field, and base points are either absent or local complete intersection isolated points.
In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in Busé and Jouanolou [2002], where implicit equations are obtained as determinants of certain graded parts of a so-called approximation complex. We detail and improve this method by providing an in-depth study of the cohomology of such a complex. In both particular cases of interest of curve and surface implicitization we also yield explicit algorithms which only involves linear algebra routines.
We show that the method of moving quadrics for implicitizing surfaces in P 3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit equation can be computed as the resultant of the first syzygies.
International audienceGiven a parametrization of a rational plane algebraic curve C, some explicit adjoint pencils on C are described in terms of determinants. Moreover, some generators of the Rees algebra associated to this parametrization are presented. The main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence
Given a parameterization of an algebraic rational curve in a projective space of arbitrary dimension, we introduce and study a new implicit representation of this curve which consists in the locus where the rank of a single matrix drops. Then, we illustrate the advantages of this representation by addressing several important problems of Computer Aided Geometric Design: The point-on-curve and inversion problems, the computation of singularities and the calculation of the intersection between two rational curves.
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