PhD Thesis: Sur La Décomposition Réelle Et Algébrique Des Systèmes Dépendant De Paramètres - Guillaume Morotz
Université Pierre et Marie Curie (2008)
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This thesis deals with parametric systems. They appear is many applications, such as robotics or camera calibration. Let S be a parametric system of polynomial equations and inequalities. We study the problem of describing the open connected sets U of the parameters’ space such that S restricted to U has a constant number of real solutions. In robotics, we describe the cuspidal configurations of planar parallel robots, important for path planing. In camera calibration, we detect explicitly the number of solutions to the Perspective-3-Points problem physically achievable. From a theoretical point of view, we analyse the problem of computing the discriminant variety of a parametric system. Under some assumptions, we show that a discriminant variety
computation can be reduced to a projection computation. In particular, we present a polynomial space algorithm to compute it. In the case of a general polynomial system, we introduce the regular decompositions, where each component is represented by a sequence of polynomials regular outside a hypersurface. To compute such decompositions, our algorithm uses mainly the saturation of polynomial ideals and seems efficient in practice. In the case of parametric systems, this representation allows us to decrease a combinatorial factors that appears in the computation of the discriminant variety. Beside, we deduce from this work a new algorithm to compute
the radical of a polynomial ideal.

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