  We develop in this paper some methods for studying the implicitization
problem for a rational map $\phi: \mathbb{P}^n \to (\mathbb{P}^1)^{n+1}$
defining a hypersurface in $(\mathbb{P}^1)^{n+1}$, based on computing the
determinant of a graded strand of a Koszul complex. We show that the classical
study of Macaulay Resultants and Koszul complexes coincides, in this case, with
the approach of approximation complexes and we study and give a geometric
interpretation for the acyclicity conditions. Under suitable hypotheses, these
techniques enable us to obtain the implicit equation, up to a power, and up to
some other extra factor. We give algebraic and geometric conditions for
determining when the computed equation defines the scheme theoretic image of
$\phi$, and, what are the extra varieties that appear. We also give some
applications to the problem of computing sparse discriminants.
