Counting Central Configurations at the Bifurcation Points

Enumeration problems for the central configurations of the Newtonian body problem are hard for in and in . These are problems in finding the numbers of classes of central configurations for all the masses in a parameter space of positive dimensions. Many results are obtained generically. That is, rigorous proofs of the counting problems only exists for parameters not at the bifurcation points. For the bifurcation points, only numerical evidences are provided due to the complexity of the problems. In this paper, we propose an algorithm that rigorously proves results on counting central configurations for all masses in one dimensional parameter spaces. Especially, we provide an approach to find all bifurcation points and count real roots at those points, known only implicitly. A spatial restricted -body problem and a planar -body problem are successfully applied by our method. All results except for the equal masses for the restricted -body problem are new and the results for the planar -body problem are new at the bifurcation points.