Finite-field Heuristics for the Center Variety of Cubic Systems

Abstract We study the global geometry of the center variety of cubic systems of planar polynomial vector fields. Using a finite–field heuristic based on a fast implementation of Frommer’s algorithm, we estimate the number of reduced components of the variety defined by the first 13 focal values. We also prove that 21 of Zoladek’s reversible and Darboux families generate reduced irreducible components. Up to codimension 8 these components account for all numerically detected ones, while from codimension 9 onward the heuristic counts are substantially larger, suggesting that the full center variety of cubic systems contains on the order of one hundred further reduced components beyond those arising from Zoladek’s families.