New equations for central configurations and generic finiteness

Abstract:

We consider the finiteness problem for central configurations of the $n$-body problem. We prove that, for $n\geq 4$, there exists a (Zariski) closed subset $B$ in the mass space $\mathbb {R}^{n}$, such that if $(m_1,\dots ,m_n) \in \mathbb {R}^n\setminus B$, 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.