Are Palmore’s “ignored estimates” on the number of planar central configurations correct?

Alain Albouy (1)

1 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides

We wish to draw attention on estimates on the number of relative equilibria in the Newtonian n-body problem that Julian Palmore published in 1975. Julian Palmore published in [12] a first estimate on the number of planar central configurations with given number of bodies n and given positive masses m 1 ,. .. , m n. If these central configurations are non-degenerate, they are at least (3n − 4)(n − 1)!/2. The word " configuration " is ambiguous. Here we consider two " configurations " which are deduced one from the other by homothety and rotation as the same configuration. But we consider that applying a reflection to a non-collinear configuration produces a distinct configuration. We also consider that the bodies are " distinguishable " : configurations which differ only in the numbering of the bodies are nevertheless considered as distinct.

Domaines: Géométrie algébrique [math.AG]

Alain Albouy. Are Palmore’s “ignored estimates” on the number of planar central configurations correct?. Qualitative Theory of Dynamical Systems, 2015, 14, pp.403-406. ⟨10.1007/s12346-015-0170-z⟩. ⟨hal-01202661⟩