Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2501.05315 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910887861813248 |
|---|---|
| author | Edelsbrunner, Herbert Fillmore, Christopher Oliveira, Gonçalo |
| author_facet | Edelsbrunner, Herbert Fillmore, Christopher Oliveira, Gonçalo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05315 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting Equilibria of the Electrostatic Potential Edelsbrunner, Herbert Fillmore, Christopher Oliveira, Gonçalo Computational Geometry Mathematical Physics Combinatorics 31B05, 52B10, 52C35, 58E05, 78A30, 78-10 I.3.5; J.2 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. |
| title | Counting Equilibria of the Electrostatic Potential |
| topic | Computational Geometry Mathematical Physics Combinatorics 31B05, 52B10, 52C35, 58E05, 78A30, 78-10 I.3.5; J.2 |
| url | https://arxiv.org/abs/2501.05315 |