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

TL;DR

This paper extends the positivity of certain Pfaffians related to collinear central configurations in the inverse problem, proving they hold for broader potentials and larger even numbers of bodies.

Contribution

It generalizes previous inequalities to potentials with log-convex derivatives and larger even numbers of bodies, up to n=14.

Findings

Pfaffian inequalities hold for potentials with log-convex derivatives.

Positivity proven for all even n ≤ 14 in the generalized case.

Extends previous results from n ≤ 6 to larger n and broader potentials.

Abstract

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 $1/r$ and $n\leq 6$). 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 $n\leq 14$).