---
OA_place: repository
OA_type: green
_id: '21931'
abstract:
- lang: eng
  text: 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.
article_number: e70163
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Christopher D
  full_name: Fillmore, Christopher D
  id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
  last_name: Fillmore
- first_name: Goncalo
  full_name: Oliveira, Goncalo
  id: 58abbde8-f455-11eb-a497-98c8fd71b905
  last_name: Oliveira
citation:
  ama: Edelsbrunner H, Fillmore CD, Oliveira G. Counting equilibria of the electrostatic
    potential. <i>Proceedings of the London Mathematical Society</i>. 2026;132(5).
    doi:<a href="https://doi.org/10.1112/plms.70163">10.1112/plms.70163</a>
  apa: Edelsbrunner, H., Fillmore, C. D., &#38; Oliveira, G. (2026). Counting equilibria
    of the electrostatic potential. <i>Proceedings of the London Mathematical Society</i>.
    Wiley. <a href="https://doi.org/10.1112/plms.70163">https://doi.org/10.1112/plms.70163</a>
  chicago: Edelsbrunner, Herbert, Christopher D Fillmore, and Goncalo Oliveira. “Counting
    Equilibria of the Electrostatic Potential.” <i>Proceedings of the London Mathematical
    Society</i>. Wiley, 2026. <a href="https://doi.org/10.1112/plms.70163">https://doi.org/10.1112/plms.70163</a>.
  ieee: H. Edelsbrunner, C. D. Fillmore, and G. Oliveira, “Counting equilibria of
    the electrostatic potential,” <i>Proceedings of the London Mathematical Society</i>,
    vol. 132, no. 5. Wiley, 2026.
  ista: Edelsbrunner H, Fillmore CD, Oliveira G. 2026. Counting equilibria of the
    electrostatic potential. Proceedings of the London Mathematical Society. 132(5),
    e70163.
  mla: Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.”
    <i>Proceedings of the London Mathematical Society</i>, vol. 132, no. 5, e70163,
    Wiley, 2026, doi:<a href="https://doi.org/10.1112/plms.70163">10.1112/plms.70163</a>.
  short: H. Edelsbrunner, C.D. Fillmore, G. Oliveira, Proceedings of the London Mathematical
    Society 132 (2026).
corr_author: '1'
date_created: 2026-05-31T22:02:13Z
date_published: 2026-05-01T00:00:00Z
date_updated: 2026-06-02T09:24:18Z
day: '01'
department:
- _id: HeEd
- _id: TaHa
doi: 10.1112/plms.70163
external_id:
  arxiv:
  - '2501.05315'
intvolume: '       132'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2501.05315
month: '05'
oa: 1
oa_version: Preprint
publication: Proceedings of the London Mathematical Society
publication_identifier:
  eissn:
  - 1460-244X
  issn:
  - 0024-6115
publication_status: published
publisher: Wiley
quality_controlled: '1'
related_material:
  record:
  - id: '21050'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: Counting equilibria of the electrostatic potential
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2026'
...