REAL ROOT COUNTING FOR CENTRAL CONFIGURATIONS

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 conguration problems for three and four bodies with specic masses, and we prove a better bound for a 4-vortex problem with three vorticities equal to one and the fourth vorticity as a variable.