In this article we give a complete global classification of the class QS ess of planar, essentially quadratic differential systems (i.e. defined by relatively prime polynomials and whose points at infinity are not all singular), according to their topological behavior in the vicinity of infinity. This class depends on 12 parameters but due to the action of the affine group and re-scaling of time, the family actually depends on five parameters. Our classification theorem (Theorem 7.1) gives us a complete dictionary connecting very simple integer-valued invariants which encode the geometry of the systems in the vicinity of infinity, with algebraic invariants and comitants which are a powerful tool for computer algebra computations helpful in the route to obtain the full topological classification of the class QS of all quadratic differential systems.

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Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert,1900], are still open for this class. Even when not dealing with limit cycles, still some problems have remained unsolved like a complete classification of different phase portraits without limit cycles. For some time it was thought (see [Coppel, 1966]) there could exist a set of algebraic functions whose signs would completely determine the phase portrait of a quadratic system. Nowadays we already know that this is not so, and that there are some analytical, nonalgebraic functions that also play a role when dealing with limit cycles and separatrix connections. However, it is possible to find out a set of algebraic functions whose signs determine the characteristics of all finite and infinite singular points. Most of the work up to now has dealt with this problem studying it using different normal forms adapted to some subclasses of quadratic systems. A general work useful for any quadratic system regardless of affine changes has only been done for the study of infinite singular points [Schlomiuk et al., 2005]. In this paper, we give a complete global classification of quadratic differential systems according to their topological behavior in the vicinity of the finite singular points. Our classification Main Theorem gives us a complete dictionary describing the local behavior of finite singular points using algebraic invariants and comitants which are a powerful tool for algebraic computations. Linking the result of this paper with the main one of [Schlomiuk et al., 2005] which uses the same algebraic invariants, it is possible to complete the algebraic classification of singular points (finite and infinite) for quadratic differential systems.

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In the topological classification of phase portraits no distinctions are made between a focus and a node and neither are they made between a strong and a weak focus or between foci of different orders.These distinction are however important in the production of limit cycles close to the foci in perturbations of the systems. The distinction between the one direction node and the two directions node, which plays a role in understanding the behavior of solution curves around the singularities at infinity, is also missing in the topological classification.In this work we introduce the notion of geometric equivalence relation of singularities which incorporates these important purely algebraic features. The geometric equivalence relation is finer than the topological one and also finer than the qualitative equivalence relation introduced in [19]. We also list all possibilities we have for singularities finite and infinite taking into consideration these finer distinctions and introduce notations for each one of them. Our long term goal is to use this finer equivalence relation to classify the quadratic family according to their different geometric configurations of singularities, finite and infinite.In this work we accomplish a first step of this larger project. We give a complete global classification, using the geometric equivalence relation, of the whole quadratic class according to the configuration of singularities at infinity of the systems. Our classification theorem is stated in terms of invariant polynomials and hence it can be applied to any family of quadratic systems with respect to any particular normal form. The theorem wegive also contains the bifurcation diagram, done in the 12-parameter space, of the geometric configurations of singularities at infinity, and this bifurcation set is algebraic in the parameter space. To determine the bifurcation diagram of configurations of singularities at infinity for any family of quadratic systems, given in any normal form, becomes thus a simple task using computer algebra calculations.1991 Mathematics Subject Classification. Primary 58K45, 34C05, 34A34.

We classify all cubic systems possessing the maximum number of invariant straight lines (real or complex) taking into account their multiplicities. We prove that there are exactly 23 topological different classes of such systems. For every class we provide the configuration of its invariant straight lines in the Poincaré disc. Moreover, every class is characterized by a set of affine invariant conditions.