  We consider the finiteness problem for central configurations of the $n-$body
problem. We prove that, for $n\geq4$, 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 $k \geq 1$, we show that the set of mutual distances associated to
a $k-$dimensional central configuration is contained in a determinantal
algebraic set.
