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Abstract:
The computation of the topological shape of a real algebraic plane curve is
usually driven by the study of the behavior of the curve around its critical
points (which includes also the singular points). In this paper we present a
new algorithm computing the topological shape of a real algebraic plane curve
whose complexity is better than the best algorithms known. This is due to the
avoiding, through a sufficiently good change of coordinates, of real root
computations on polynomials with coefficients in a simple real algebraic
extension of $\mathbb{Q}$ to deal with the critical points of the considered
curve. In fact, one of the main features of this algorithm is that its
complexity is dominated by the characterization of the real roots of the
discriminant of the polynomial defining the considered curve.
