  In this paper, we investigate the structure of the saturation of ideals
generated by sparse homogeneous polynomials over a projective toric variety $X$
with respect to the irrelevant ideal of $X$. As our main results, we establish
a duality property and make it explicit by introducing toric Sylvester forms,
under a certain positivity assumption on $X$. In particular, we prove that
toric Sylvester forms yield bases of some graded components of
$I^{\text{sat}}/I$, where $I$ denotes an ideal generated by $n+1$ generic
forms, $n$ is the dimension of $X$ and $I^{\text{sat}}$ the saturation of $I$
with respect to the irrelevant ideal of the Cox ring of $X$. Then, to
illustrate the relevance of toric Sylvester forms we provide three consequences
in elimination theory over smooth toric varieties: (1) we introduce a new
family of elimination matrices that can be used to solve sparse polynomial
systems by means of linear algebra methods, including overdetermined polynomial
systems; (2) by incorporating toric Sylvester forms to the classical Koszul
complex associated to a polynomial system, we obtain new expressions of the
sparse resultant as a determinant of a complex; (3) we explote a new formula
for computing toric residues of the product of two forms.

Algebraic Geometry;Commutative Algebra