Complex algebraic plane curves via the Poincare-Hopf formula - I. Parametric lines

We present a new and very efficient approach to study topology of algebraic curves in C-2. It relies on using the Poincare-Hopf formula, applied to a suitable Hamiltonian vector field, to estimate the number of double points and the Milnor numbers of singular points of the curve and on considering finite-dimensional spaces of curves with given asymptotics at infinity. We apply this method to classification of parametric lines with one self-intersection.