Pfaffians and the inverse problem for collinear central configurations

Abstract

We consider, after Albouy-Moeckel, the inverse problem for collinear central configurations: given a collinear configuration of n bodies, find positive masses which make it central. We give some new estimates concerning the positivity of Albouy-Moeckel pfaffians: we show that for any homogeneity α and n ≤ 6 or n ≤ 10 and α = 1 (computerassisted) the pfaffians are positive. Moreover, for the inverse problem with positive masses, we show that for any homogeneity and n ≥ 4 there are explicit regions of the configuration space without solutions of the inverse problem.