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 $\alpha$ and $n\le 6$ or $n\le 10$ and $\alpha =1$ (computer-assisted) the pfaffians are positive. Moreover, for the inverse problem with positive masses, we show that for any homogeneity and $n\ge 4$ there are explicit regions of the configuration space without solutions of the inverse problem.