The implicitization problem for

We develop in this paper methods for studying the implicitization problem for a rational map defining a hypersurface in (P1)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 extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of ϕ, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants.