Finiteness of Newtonian equilibrium points

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $n\in\mathbb N$, let $x_1,\ldots,x_n\in\mathbb R^3$ be distinct points, and let $a_1,\ldots,a_n>0$. Repeated locations can be combined by adding their positive charges. On $\Omega=\mathbb R^3\setminus\{x_1,\ldots,x_n\}$ define \[ u(x)=\sum_{k=1}^{n}\frac{a_k}{\lVert x-x_k\rVert},\qquad \operatorname{Crit}(u)=\{x\in\Omega:\nabla u(x)=0\}. \] In coordinates this is the solution set of \[ \sum_{k=1}^{n}a_k\frac{x-x_k}{\lVert x-x_k\rVert^3}=0. \] The locations of the charges are excluded from the critical set.

Statement recorded in the supplied source.

Is $\operatorname{Crit}(u)$ finite for every such finite positive-charge configuration? No generic-position or nondegeneracy assumption is imposed. In particular, a bound for isolated equilibria alone does not establish this assertion: non-isolated equilibria must also be excluded. This finiteness question is weaker than Maxwell's proposed $(n-1)^2$ bound. [S2]

Short English statement

Must finitely many positive point charges in three-dimensional space have only finitely many equilibrium points, even for nongeneric configurations?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3700035 (AMR-036-0035). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Herbert Edelsbrunner, Christopher Fillmore and Gonçalo Oliveira, Counting equilibria of the electrostatic potential (2025), §1, equation (1), Conjecture 1 and Result 1, pp.1–3. https://arxiv.org/abs/2501.05315.