The codimension of the phase portraits for degenerate quadratic differential systems

In this paper we present a complete study of degenerate quadratic differential systems, i.e. the polynomials from right-hand sides of these systems are not co-prime. We give the complete set of their phase portraits together with the necessary and sufficient conditions for the realization of each one of them. These conditions are given in using invariant polynomials and we present here the "bifurcation" diagram directly in the space R12 of the whole set of the parameters of the quadratic systems. This paper is part of a project whose ultimate goal is the complete classification of all topologically distinct phase portraits of quadratic systems modulo limit cycles. We also provide a label for each phase portrait inside the global codification related to the global configurations of singularities and their topological codimensions.