Implicitizing rational hypersurfaces using approximation complexes

We describe an algorithm for implicitizing rational hypersurfaces with at most a finite number of base points, based on a technique described in Busé, Laurent, Jouanolou, Jean-Pierre [2003. On the closed image of a rational map and the implicitization problem. J. Algebra 265, 312-357], where implicit equations are obtained as determinants of certain graded parts of an 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 present algorithms which involve only linear algebra routines.