On the irreducibility of multivariate subresultants

Laurent Busé 1 Carlos d'Andrea 2
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : Let $P_1,\ldots,P_n$ be generic homogeneous polynomials in $n$ variables of degrees $d_1,\ldots,d_n$ respectively. We prove that if $\nu$ is an integer satisfying ${\sum_{i=1}^n d_i}-n+1-\min\{d_i\}<\nu,$ then all multivariate subresultants associated to the family $P_1,\ldots,P_n$ in degree $\nu$ are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of $\binom{\rho-\nu +n-1}{n-1}$ smooth isolated points in $\PP^{n-1}.$
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Laurent Busé, Carlos d'Andrea. On the irreducibility of multivariate subresultants. Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2004, 338 (4), pp.287--290. ⟨inria-00098679⟩

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