Lower and upper bounds on the number of planar central configurations [ Back ]

14:30 to 15:30
CRM - Auditori (C1/034)
Alain Albouy
Observatoire de Paris


The planar central configurations are configurations of a finite number of gravitating points in a relative equilibrium.  They are obtained by solving a system of algebraic equations. This is one of the most natural systems having configurations as roots. By looking at the set of solutions we observe very simple facts, which we are unable to prove. For example, whatever be the masses, there are at most 50 planar central configurations of 4 bodies. We will analyse results by Marshall Hampton, Rick Moeckel, Carles Simó and Zhihong Xia which point toward an interesting conjecture about a general lower bound.
This conjecture is indeed due to Palmore, but is strengthened by comparing it to Xia's and Simó's papers. I will use results by Moeckel and Tien (see Moeckel's manuscript notes, p. 81), and an unpublished work by Joe Fayad.