Central configurations and polynomial elimination

par Dr Alain Albouy (ASD)

jeudi 11 sept. 2025, 11:00 → 11:55 Europe/Paris

Denisse (Paris)

The n-body problem of celestial mechanics has simple motions called the relative equilibria. The configuration of the bodies in such a motion is called central.

In 1996 I proved by using polynomial elimination on Maple that there are only 3 types of symmetric non-collinear central configurations of 4 equal masses. My preceding paper had proved the symmetry of these planar central configurations.

In 2019 another computer assisted method, based on interval arithmetic, reproved these results and the similar ones for 5, 6 and 7 bodies with equal masses (Moczurad, Zgliczynski).

These methods are less efficient when dealing with equations with parameters. Here the important parameters are the masses of the bodies.
Faugère, Kotsireas and Lazard studied in particular the symmetric case with 4 bodies, using polynomial elimination, showing a lot of bifurcations lines where the number of solutions changes, many of them out of the domain of positive masses. The thesis of Leandro (2003) under the direction of Moeckel gave a simple picture. A recent publication by Roberts (2025) reconsiders the question. I will try to present these works.