In this talk we will discuss two types of algebraic methods to solve the implicitization problem, as well as their advantages and disadvantages.
We will begin with the resultant-based methods. It is well-known how resultants can be used to implicitize a rational surface in the absence of base points. We will show that base points can be taken into account using what is called residual resultant.
Then we will discuss the syzygy-based methods. Such methods were introduced by Sederberg under the name "moving surfaces" method. After recalling some known results on this method we will present a commutative algebra framework for discussing this method (the so-called approximation complexes). This approach gives new techniques to solve the implicitization problem.
Finally, if time remains we will discuss on another possibility for representing algebraic surfaces as a parameterized family of implicit curves (we will call such a representation a semi-implicit representation). By the way we will present a needed tool, a projection operator (in some sense a resultant), to manipulate and convert them.