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We develop in this paper some methods for studying the implicitizationproblem 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 thedeterminant of a graded strand of a Koszul complex. We show that the classicalstudy of Macaulay Resultants and Koszul complexes coincides, in this case, withthe approach of approximation complexes and we study and give a geometricinterpretation for the acyclicity conditions. Under suitable hypotheses, thesetechniques enable us to obtain the implicit equation, up to a power, and up tosome other extra factor. We give algebraic and geometric conditions fordetermining when the computed equation defines the scheme theoretic image of$\phi$, and, what are the extra varieties that appear. We also give someapplications to the problem of computing sparse discriminants.

The implicitization problem for $\phi: P^n --> (P^1)^{n+1}$