Implicitization using approximation complexes

We present a method for the implicitization problem that goes back to the work of Sederberg and Chen. The formalism we use, with approximation complexes as a key ingredient, is due to Jean-Pierre Jouanolou and was explained in details in this context in his joint work with Laurent Buse. Most of this note is dedicated to presenting the method, the geometric ideas behind it and the tools from commutative algebra that are needed. In the last section, we give the most advanced results we know related to this approach.