  Motivated by the interest in computing explicit formulas for resultants and
discriminants initiated by B\'ezout, Cayley and Sylvester in the eighteenth and
nineteenth centuries, and emphasized in the latest years due to the increase of
computing power, we focus on the implicitization of hypersurfaces in several
contexts. Our approach is based on the use of linear syzygies by means of
approximation complexes, following [Bus\'e Jouanolou 03], where they develop
the theory for a rational map $f:P^{n-1}\dashrightarrow P^n$. Approximation
complexes were first introduced by Herzog, Simis and Vasconcelos in [Herzog
Simis Vasconcelos 82] almost 30 years ago. The main obstruction for this
approximation complex-based method comes from the bad behavior of the base
locus of $f$. Thus, it is natural to try different compatifications of
$A^{n-1}$, that are better suited to the map $f$, in order to avoid unwanted
base points. With this purpose, in this thesis we study toric compactifications
$T$ for $A^{n-1}$. We provide resolutions $Z.$ for $Sym_I(A)$, such that
$\det((Z.)_\nu)$ gives a multiple of the implicit equation, for a graded strand
$\nu\gg 0$. Precisely, we give specific bounds $\nu$ on all these settings
which depend on the regularity of $\SIA$. Starting from the homogeneous
structure of the Cox ring of a toric variety, graded by the divisor class group
of $T$, we give a general definition of Castelnuovo-Mumford regularity for a
polynomial ring $R$ over a commutative ring $k$, graded by a finitely generated
abelian group $G$, in terms of the support of some local cohomology modules. As
in the standard case, for a $G$-graded $R$-module $M$ and an homogeneous ideal
$B$ of $R$, we relate the support of $H_B^i(M)$ with the support of
$Tor_j^R(M,k)$.
