Counting equilibria of the electrostatic potential

Edelsbrunner, Herbert; Fillmore, Christopher D; Olivera, Gonçalo

Counting equilibria of the electrostatic potential

Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv, 10.48550/ARXIV.2501.05315.

Download (ext.)

https://doi.org/10.48550/arXiv.2501.05315 [Preprint]

DOI

10.48550/ARXIV.2501.05315

Preprint | Draft | English

Author

Edelsbrunner, Herbert^ISTA ; Fillmore, Christopher D^ISTA; Olivera, Gonçalo

Corresponding author has ISTA affiliation

Department

Edelsbrunner Group

Abstract

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 (non-optimal) 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 a conjecture of 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 smaller 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.

Publishing Year

2025

Date Published

2025-03-20

Journal Title

arXiv

IST-REx-ID

21050

Cite this

Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv. doi:10.48550/ARXIV.2501.05315

Edelsbrunner, H., Fillmore, C. D., & Olivera, G. (n.d.). Counting equilibria of the electrostatic potential. arXiv. https://doi.org/10.48550/ARXIV.2501.05315

Edelsbrunner, Herbert, Christopher D Fillmore, and Gonçalo Olivera. “Counting Equilibria of the Electrostatic Potential.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2501.05315.

H. Edelsbrunner, C. D. Fillmore, and G. Olivera, “Counting equilibria of the electrostatic potential,” arXiv. .

Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv, 10.48550/ARXIV.2501.05315.

Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” ArXiv, doi:10.48550/ARXIV.2501.05315.

All files available under the following license(s):

Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):