Complex algebraic plane curves via the Poincaré-Hopf formula. I. Parametric lines
There do not exist simple rules to determine the topology of a plane algebraic curve, whether given in implicit or parametric form. This paper presents a new approach to this problem. The authors consider parametric polynomial curves; they use a Poincaré-Hopf formula applied to a suitable Hamiltonian vector field to calculate the number of double points and the Milnor numbers of singular points of the curve and study finite-dimensional spaces of curves with given asymptotics at infinity. This method is then applied to classify parametric lines (i.e. plane curves \(\xi: {\mathbb C} \rightarrow {\mathbb C}^2, \xi(t)=((\phi(t), \phi(t)))\) where \((\phi, \phi)\) are polynomials) with one self-intersection.
scientific article; zbMATH DE number 5366196