The implicitization problem for φ{symbol} : Pn (P1)n + 1 - Botbol, N. (2009)
We develop in this paper methods for studying the implicitization problem for a rational map φ{symbol} : Pn (P1)n + 1 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{symbol}, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants. © 2009 Elsevier Inc. All rights reserved.