Inverse problem of central configurations and singular curve in the collinear 4-body problem

In this paper, we consider the inverse problem of central configurations of n-body problem. For a given ({q=(q_1, q_2, ldots, q_n)in ({bf R}^d)^n}), let S(q) be the admissible set of masses denoted ({ S(q)={ m=(m_1,m_2, ldots, m_n)| m_i in {bf R}^+, q}) is a central configuration for m}. For a given ({min S(q)}), let S _m (q) be the permutational admissible set about m = (m ₁, m ₂, . . . , m _n ) denoted

$S_m(q)={m^prime | m^primein S(q),m^prime not=m , {rm and} , m^prime,{rm is, a, permutation, of }, m }.$

The main discovery in this paper is the existence of a singular curve ({bar{Gamma}_{31}}) on which S _m (q) is a nonempty set for some m in the collinear four-body problem. ({bar{Gamma}_{31}}) is explicitly constructed by a polynomial in two variables. We proved:

1. (1)
   If ({min S(q)}), then either ^# S _m (q) = 0 or ^# S _m (q) = 1.
    
2. (2)
   ^#S _m (q) = 1 only in the following cases:

   1. (i)
      If s = t, then S _m (q) = {(m ₄, m ₃, m ₂, m ₁)}.
       
   2. (ii)
      If ({(s,t)in bar{Gamma}_{31}setminus {(bar{s},bar{s})}}), then either S _m (q) = {(m ₂, m ₄, m ₁, m ₃)} or S _m (q) = {(m ₃, m ₁, m ₄, m ₂)}.