New equations for central configurations and generic finiteness

We consider the finiteness problem for central configurations of the $n-$body problem. We prove that, for $n\ge 4$, there exists a (Zariski) closed subset $B$ in the mass space ${\mathbb{R}}^n$, such that if $(m_1,…,m_n) \in $\mathbb{R}$^n\setminus B$, then there is a finite number of corresponding classes of $(n-2)-$dimensional central configurations for potential associated to a semi-integer exponent. Also, we obtain trilinear homogeneous polynomial equations of degree $3$ for central configurations of fixed dimension and, for each integer $k\ge 1$, we show that the set of mutual distances associated to a $k-$dimensional central configuration is contained in a determinantal algebraic set.