Finiteness and Bifurcations of some Symmetrical Classes of Central Configurations

We consider here the classical question of finiteness (see Smale [13] and Wintner [15]) – given n point masses, is the corresponding number of central configurations finite? We prove finiteness for a particular family of d-dimensional symmetrical configurations of d+2 point masses. Also, we study the bifurcations of these configurations and provide the exact number of central configurations when d=2, 3. All our results stem from the application of a new method for studying symmetrical classes of central configurations, which is presented in this work.