Pfaffians and the inverse problem for collinear central configurations

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 (computer assisted) 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.