Abstract:

We will continue and finish this series of talks on central configurations for homogeneous potentials with integer exponents. We present a polynomial parametrization for central configurations of dimension n-k. Also, we give a criterion for the dimension of an arbitrary central configuration in terms of the Cayley-Menger determinant. Finally, by using the Jacobian Criterion from Algebraic Geometry, we prove generic finiteness with respect to the masses for (n-2)-dimensional central configurations in the n-body problem under homogeneous potentials with integer exponents.