  We unveil in concrete terms the general machinery of the syzygy-based
algorithms for the implicitization of rational surfaces in terms of the
monomials in the polynomials defining the parametrization, following and
expanding our joint article with M. Dohm. These algebraic techniques, based on
the theory of approximation complexes due to J. Herzog, A, Simis and W.
Vasconcelos, were introduced for the implicitization problem by J.-P.
Jouanolou, L. Bus\'e, and M. Chardin. Their work was inspired by the practical
method of moving curves, proposed by T. Sederberg and F. Chen, translated into
the language of syzygies by D. Cox. Our aim is to express the theoretical
results and resulting algorithms into very concrete terms, avoiding the use of
the advanced homological commutative algebra tools which are needed for their
proofs.
