Avoiding the general position condition whencomputing the topology of a real algebraic planecurve defined implicitly

Abstract

The problem of computing the topology of curves has re-ceived special attention from both Computer Aided Geometric Designand Symbolic Computation. It is well known that the general positioncondition simplifies the computation of the topology of a real algebraicplane curve defined implicitly since, under this assumption, singularpoints can be presented in a very convenient way for that purpose. Herewe will show how the topology of cubic, quartic and quintic plane curvescan be computed in the same manner even if the curve is not in gen-eral position, avoiding thus coordinate changes. This will be possibleby applying new formulae, derived from subresultants, which describemultiple roots of univariate polynomials as rational functions of the con-sidered polynomial coefficients. We will also characterize those higherdegree curves where this approach can be used and use this technique todescribe the curve arising when intersecting two ellipsoids.