On Certain Pfaffians Connected with the Inverse Problem for Collinear Central Configurations

A. Albouy and R. Moeckel in 2000 found some interesting inequalities related to the inverse problem for collinear (Moulton) central configurations: the Pfaffian of a certain matrix is positive since all coefficients of some polynomials are positive, for the Newtonian (interaction potential $1\mathrm{/}r$ and $n\le 6$). They conjectured that for all $n$ such Pfaffians, for the Newtonian case, are positive. In this article we analyze further the problem, and we prove that such inequalities hold true in more general cases (potentials with log-convex derivative, such as those with homogeneity parameters $\alpha >0$, for all even $n\le 14$).