  In this paper we study the equations of the elimination ideal associated with
$n+1$ generic multihomogeneous polynomials defined over a product of projective
spaces of dimension $n$. We first prove a duality property and then make this
duality explicit by introducing multigraded Sylvester forms. These results
provide a partial generalization of similar properties that are known in the
setting of homogeneous polynomial systems defined over a single projective
space. As an important consequence, we derive a new family of elimination
matrices that can be used for solving zero-dimensional multiprojective
polynomial systems by means of linear algebra methods.
