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We present a new and very eficient approach to study topology of algebraic curves in C2. 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.

Complex algebraic plane curves via the Poincaré–Hopf formula, I: Parametric lines