Codimension in planar polynomial differential systems

The topological classification of large families of polynomial differential systems remains an extremely difficult problem. Even for the simplest nonlinear case, the quadratic systems, this problem is still open. In recent years however, significant progress has been achieved on a related challenge: the topological classification modulo limit cycles of the family QS of quadratic differential systems. This progress was enabled by introducing new tools, such as the concepts of the global geometrical and topological configurations of singularities, and by first classifying QS with respect to these notions.I n this work, we extend Sotomayor's notion of codimension, originally based on topological equivalence, so as to allow for other equivalence relations including the geometric one. We provide a rigorous, improved definition of codimension that could be used efficiently for obtaining the classification of the phase portraits of QS modulo limit cycles, and not just for small codimensions as it usually occurs in the present literature. This new definition of codimension can be extended to arbitrary polynomial differential systems, and other differential systems. In this work we apply the new concept of codimension by assigning a codimension to each one of the 207 global topological configurations of singularities of systems in QS. This tool is of great help for assigning codimensions even to phase portraits modulo limit cycles. This concept is a potent tool in the topological classification problem modulo limit cycles in QS. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/28/abstr.html