Topology of Plane Algebraic Curves

Topology of Plane Algebraic Curves: the Algebraic Approach

We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

Certified Rational Parametric Approximation of Real Algebraic Space Curves with Local Generic Position Method

In this paper, an algorithm to determine the topology of an algebraic space curve and to compute a certified G rational parametric approximation of the algebraic space curve is given by extending the local generic position method for solving zero dimensional polynomial equation systems to the case of dimension one. By certified, we mean the approximation curve and the original curve have the sa...

On the Topology of Real Algebraic Plane Curves

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylind...

Analyzing the Topology Types arising in a Family of Algebraic Curves Depending On Two Parameters

Given the implicit equation F (x, y, t, s) of a family of algebraic plane curves depending on the parameters t, s, we provide an algorithm for studying the topology types arising in the family. For this purpose, the algorithm computes a finite partition of the parameter space so that the topology type of the family stays invariant over each element of the partition. The ideas contained in the p...

Solving bivariate algebraic systems and topology of plane curves. (Résolution de systèmes bivariés et topologie de courbes planes)