Resultant over the residual of a complete intersection

Laurent Busé 1, 2 Mohamed Elkadi 1 Bernard Mourrain 1
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal (F:G) and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples.
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Laurent Busé, Mohamed Elkadi, Bernard Mourrain. Resultant over the residual of a complete intersection. Journal of Pure and Applied Algebra, Elsevier, 2001, Effective methods in algebraic geometry (Bath, 2000), 164 (1-2), pp.35--57. ⟨inria-00099487⟩

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