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A new mathematical example shows that a conjecture attributed to J. C. Maxwell about the number of equilibria of an electric field is false. The authors construct five positive point charges in three‑dimensional space whose electrostatic potential has at least 24 non‑degenerate critical points. For five charges the Maxwell conjecture would predict at most (5−1)^2 = 16 such points, so this example violates that bound.
What the paper studies are the equilibria of the electric field. Equilibria are points where the field vanishes, equivalently critical points of the electrostatic potential. “Non‑degenerate” here means isolated equilibria whose second‑derivative matrix (the Hessian) is invertible, so they are robust to small changes. The authors start with three unit charges placed at the corners of an equilateral triangle. That triangular configuration has four equilibria: one at the center and three nearer the edges. They then add two very small positive charges on the triangle’s symmetry axis, one slightly above and one slightly below the plane, making a shallow triangular bipyramid. In this configuration the three edge equilibria persist, while the central equilibrium splits into a family of 21 new non‑degenerate equilibria, giving at least 24 in total.
The construction is proved using elementary analytic tools and careful expansions. The authors rescale the coordinates around the triangle center and write a deformed potential Φε that captures the leading behavior as the small axial charges shrink. They compute the limit polynomial Φ0 explicitly and show it has exactly 21 non‑degenerate critical points by writing it in cylindrical coordinates and checking the derivatives and second derivatives. The implicit function theorem then shows these 21 critical points persist for sufficiently small axial charges. The paper gives explicit formulas for the small axial charge strengths and reports computer algebra checks (Mathematica and Maple) used to verify the computations and to produce visualizations.
The authors also address the possibility of other degenerate equilibria far from the origin. They use a transversality argument: by a tiny perturbation of the charges one obtains a Morse potential (a function with only non‑degenerate critical points) for almost every nearby choice of charge strengths. Because all charges are positive, any critical point must lie in the convex hull of the five charge locations, so there are only finitely many. The paper notes that the initial idea for the construction was suggested by a large language model, but that the authors verified and wrote the mathematics themselves.
Why this matters and what it does not do. The example gives a concrete counterexample to Maxwell’s conjecture and shows that the naive (n−1)^2 bound can fail already for n=5. The authors further sketch an iterative version of the construction that, for every nonnegative integer m, produces configurations of 3+2m positive charges with at least 4+20m non‑degenerate critical points, yielding an asymptotic critical‑point‑to‑charge ratio of about 10. Important caveats are that the result is an existence proof based on a carefully chosen, small‑parameter construction. It does not propose a new universal upper bound for all configurations, and the argument depends on local expansions and perturbation theorems that apply in the small‑charge limit.