Real Root-Counting Techniques for Central Configuration Problems

The n-body problem seeks to predict the motions of n masses that attract each other according to the Newtonian gravitational law. A central configuration is an arrangement of masses for which the accelerations are all parallel to the displacements from the center of mass with equal proportionality constants. Central configurations are important because they can be used to produce the only known explicit solutions of the n-body problem. One of the major underlying questions is whether there are only finitely many ways to arrange n given masses to form a central configuration (up to translation, rotation, and scaling). In terms of the mutual distances, central configurations are described as solutions of systems of polynomial equations. Hampton and Moeckel showed that the set of all complex solutions for the equations for four masses is finite. However, being that mutual distances must be positive reals, not all complex solutions of the equations are physically relevant.

We use methods of algebraic geometry to count the number of real solutions of systems of polynomial equations in a given region of Euclidean space, such as the positive orthant consisting of points with all positive coordinates. These root-counting methods yield the exact number of positive real solutions and do not rely on sensitive numerical techniques. We apply these methods to several central configuration problems for three and four masses.