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Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$ (resp. $P(x,y,z)=\\frac{\\partial P}{\\partial z}(x,y,z)=0$), and generically its singularities are nodes (resp. nodes and ordinary cusps). State-of-the-art numerical algorithms compute the topology of smooth curves but usually fail to certify the topology of singular ones. 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