The goal of this paper is to present applications of symbolic calculations and polynomial invariants to the problem of classifying planar polynomial systems of differential equations. For these applications, we use some previously defined, and some new polynomial invariants. This is part of a much larger work by the authors together with J.C. Artes and J. Llibre which is in progress. We show here how polynomial invariants and their symbolic calculations are instrumental in obtaining the bifurcation diagram of the global configurations of singularities (finite and infinite), of quadratic differential systems having a unique simple finite singularity. This bifurcation diagram is given in the twelve-dimensional space of the coefficients of the systems, and the bifurcation points form an algebraic set. The classification of singularities is done using the notion of geometric equivalence relation of configurations of singularities, which is finer than the topological equivalence. The bifurcation diagram is expressed in terms of polynomial invariants. The results can, therefore, be applied to any family of quadratic systems, given in any normal form. Determining the configurations of singularities for any family of quadratic systems thus becomes a simple task using computer symbolic calculations.
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