Finiteness of central configurations

Remarks

This is the sixth problem of Smale's 1998 list, where it is stated for relative equilibria of the n-body problem. The question was asked earlier by Chazy and by Wintner. For $n = 3$ it is classical: there are exactly five classes, the equilateral triangle of Lagrange and the three collinear configurations of Euler. Moulton showed that for every $n$ there are exactly $n!/2$ collinear classes. Hampton and Moeckel proved finiteness for $n = 4$ in 2006, with a computer-assisted algebraic argument. Albouy and Kaloshin proved it for $n = 5$ in 2012, except possibly for masses in a codimension-two subvariety of the mass space. The question is open for $n \ge 6$. The positivity of the masses matters: Roberts found a continuum of relative equilibria in a five-body problem with one negative mass.