Counting Equilibria of the Electrostatic Potential

TL;DR

This work advances the study of equilibria for the electrostatic potential generated by $n$ positive point charges in $ obreak{\mathbb{R}}^3$. By encoding the critical-point equations as a polynomial system and applying affine Bézout, it establishes the best-known general upper bound on isolated equilibria as $2^n(3n-2)^3$, and analogously for even $p$ in the modified potentials $V_p$. It also constructs a counterexample to the conjecture that the number of equilibria cannot exceed those of the Euclidean distance function, using a truncation-octahedron configuration, and demonstrates configurations achieving high equilibria-to-vertex ratios (notably via iterative anti-prisms). Additionally, the paper analyzes regular and symmetric configurations through Morse theory and local-homology, and investigates the relationship with Voronoi tessellations and slices, offering both bounds and conjectures for refined limits. Overall, it refines the landscape of Maxwell's bound, reveals surprising counterexamples, and maps how geometry and symmetry influence equilibrium counts and their potential maxima.

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.

Figures (7)

• Figure 1: Upper row, from left to right: the binary functions on the unit $2$-sphere for a non-critical point, a minimum, a $1$-saddle, a $2$-saddle, and a maximum. Lower row, from left to right: the binary functions for the centers of the tetrahedron, cube, octahedron, dodecahedron, and isocahedron (these are degenerate equilibria and so are neither $1$-saddles nor $2$-saddles).
• Figure 2: From left to right: the hexagonal, pentagonal, square anti-prisms with the heights chosen to maximize the number of equilibria. The ratios of equilibria over vertices are $37/12 < 31/10 < 25/8$, respectively. Observe how a ring of alternating $1$- and $2$-saddles gets successively more concentrated around the center.
• Figure 3: Cut-away views of three level sets of $V_{1}$ (upper row) and three level sets of $V_{1.3}$ (lower row) defined by point sources at the vertices of the octahedron. From left to right: the values are chosen slightly less than, equal to, and slightly greater than the potential at the center of the octahedron. Removing the front of the surface reveals some of the complication at the center, which for $V_{1}$ is a degenerate equilibrium; compare with Figure \ref{['fig:binary_functions']}, but for $V_{1.3}$ is a minimum with a single point in the level set at the center.
• Figure 4: Equilibria of the electrostatic potential generated by unit point charges at the vertices of the truncated octahedron. In total there are $36$light blue$2$-saddles, $18$dark red$1$-saddles, and the degenerate equilibrium at the center. For better visualization, we split these equilibria into two groups, with one $1$-saddle and four $2$-saddles near each of the six squares displayed in the left panel, and one $1$-saddle as well as one $2$-saddle for each of the twelve edges shared by two hexagons in the right panel.
• Figure 5: The equilibria of $V_2$ generated by unit point charges at the vertices of the truncated octahedron. Compared with $V_1$, we note a drastically reduced number of $1$-saddles and a minimum at the origin; see Figure \ref{['fig:BCC-counterexample']} where we used two copies of the solid to show all equilibria. While $p = 2$ is still not large, the equilibria are already close to the barycenters of the facets and edges of the solid.

Theorems & Definitions (35)

• Conjecture 1: Maxwell in Max54
• Conjecture 2: Conjecture 1.8 (a) in GNS07
• Conjecture 3: Conjecture 1.9 in GNS07
• Lemma 1
• proof
• Theorem 1: Affine version of Bezout's theorem
• Lemma 2
• proof
• Remark 1
• Proposition 1