The implicitization problem for |
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Authors: | Nicols Botbol |
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Institution: | aDepartamento de Matemática, FCEN, Universidad de Buenos Aires, Argentina;bInstitut de Mathématiques de Jussieu, Université de P. et M. Curie, Paris VI, France |
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Abstract: | We develop in this paper methods for studying the implicitization problem for a rational map defining a hypersurface in , 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 , and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants. |
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Keywords: | Elimination theory Rational map Syzygy Approximation complex Koszul complex Implicitization |
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