Counting equilibria of the electrostatic potential

article Counting equilibria of the electrostatic potential published yes Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Christopher D Fillmore author 35638A5C-AAC7-11E9-B0BF-5503E6697425 Goncalo Oliveira author 58abbde8-f455-11eb-a497-98c8fd71b905 HeEd department TaHa department In 1873, James C. Maxwell conjectured that the electric field generated by n point charges in generic position has at most (n-1)^2 isolated zeroes. The first (nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to Conjecture 1.8 by Gabrielov, Novikov, and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find lower bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day. Wiley2026 eng Proceedings of the London Mathematical Society 0024-6115 1460-244X 2501.0531510.1112/plms.70163 1325 https://research-explorer.ista.ac.at/record/21050 Edelsbrunner H, Fillmore CD, Oliveira G. Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. 2026;132(5). doi:<a href="https://doi.org/10.1112/plms.70163">10.1112/plms.70163</a> Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” Proceedings of the London Mathematical Society, vol. 132, no. 5, e70163, Wiley, 2026, doi:<a href="https://doi.org/10.1112/plms.70163">10.1112/plms.70163</a>. H. Edelsbrunner, C.D. Fillmore, G. Oliveira, Proceedings of the London Mathematical Society 132 (2026). Edelsbrunner, Herbert, Christopher D Fillmore, and Goncalo Oliveira. “Counting Equilibria of the Electrostatic Potential.” Proceedings of the London Mathematical Society. Wiley, 2026. <a href="https://doi.org/10.1112/plms.70163">https://doi.org/10.1112/plms.70163</a>. Edelsbrunner, H., Fillmore, C. D., & Oliveira, G. (2026). Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. Wiley. <a href="https://doi.org/10.1112/plms.70163">https://doi.org/10.1112/plms.70163</a> Edelsbrunner H, Fillmore CD, Oliveira G. 2026. Counting equilibria of the electrostatic potential. Proceedings of the London Mathematical Society. 132(5), e70163. H. Edelsbrunner, C. D. Fillmore, and G. Oliveira, “Counting equilibria of the electrostatic potential,” Proceedings of the London Mathematical Society, vol. 132, no. 5. Wiley, 2026. 219312026-05-31T22:02:13Z2026-06-02T09:24:18Z