Maxwell’s quadratic upper-bound conjecture

Prove that the electrostatic potential generated by n point charges in three-dimensional Euclidean space has at most (n−1)^2 equilibria.

Background

The paper studies the number of equilibria, or critical points, of the electrostatic potential generated by finitely many point charges in R^3. A Morse-theoretic argument gives a general lower bound of n−1 equilibria for generic configurations, while Maxwell’s nineteenth-century conjecture proposes a quadratic upper bound.

The paper emphasizes that the conjectured bound is attained for n=3 but that configurations attaining it are not known for n>3. The authors instead construct configurations with a ratio of equilibria approaching 25/7, without resolving Maxwell’s proposed upper bound.