Représentations matricielles en théorie de l'élimination et applications à la géométrie

Laurent Busé 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 habilitation thesis, a matrix-based approach of elimination theory is described and illustrated through applications in algebraic modeling. This matrix-based approach allows to build a bridge between geometry and numerical linear algebra, so that some geometric problems can be given to the powerful numerical linear algebra tools. The first chapter deals with matrix-based implicit representations of rational hypersurfaces in a projective space and a new method to address the computation of the intersection locus between a rational curve and a rational surface is exposed. The second chapter contains a matrix-based implicit representation of a rational curve in a projective space of arbitrary dimension. Then, the usefulness of such a representation is illustrated with an algorithm to treat the intersection problem between two rational curves. In last chapter, a matrix-based approach to Ruppert's irreducibility criterion is given and used to improve the counting of reducible fibers in a pencil of algebraic hypersurfaces whose general member is irreducible.
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Habilitation à diriger des recherches
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Laurent Busé. Représentations matricielles en théorie de l'élimination et applications à la géométrie. Mathématiques [math]. Université Nice Sophia Antipolis, 2011. ⟨tel-00593603⟩

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