Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.05315

Counting Equilibria of the Electrostatic Potential

Herbert Edelsbrunner·Christopher Fillmore·Gonçalo Oliveira
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.CGmath-phmath.COmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Herbert Edelsbrunner, Christopher Fillmore, Gonçalo Oliveira. 2025-03-20. Counting Equilibria of the Electrostatic Potential. https://arxiv.org/abs/2501.05315

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Stretch Factor of Planar Delaunay Triangulations Is Less Than 1.65

Delaunay triangulations are a fundamental class of plane spanners, and determining their worst-case stretch factor has been a longstanding problem in computational geometry. We prove an upper bound of \(1.65\), improving the bound of \(1.998\) due to Xia (2011) and reducing the gap to the known lower bound of \(1.5932\) by a factor of more than seven. Our proof works with the chains of circumdisks introduced by Xia, along which a path between two sites is assembled disk by disk. Xia measures such a path against a quantity attached to the whole chain, and because that quantity is not additive, his induction has to be carried alongside a separate global estimate. Our main idea is to measure the path against the progress it makes along the segment joining the two sites. This quantity is additive, so the bound becomes a Bellman recursion that forgets all but one number about the disks already passed, and we show that the bound holds if and only if a potential on the current state satisfies three local inequalities. The smallest feasible potential is the value function of that recursion, so searching for a potential becomes the problem of fitting this value function from above. The geometry of the disks reduces the fit to a linear program over functions of one variable, in which a GPT-based multi-agent system that we developed found a feasible point, certified in exact arithmetic.

cs.CG↗
arXiv2026

Fast Persistent Homology Computation for Functions on $\mathbb{R}$

0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the connected components and we get the persistence diagram in time $O(nα(n))$. The running time is thus usually dominated by sorting the edges in $Θ(n\log(n))$. A little-known fact is that, in the particularly simple case of studying the sublevel sets of a piecewise-linear function on $\mathbb{R}$ or $\mathbb{S}^1$, persistence can actually be computed in linear time. This note presents a simple algorithm that achieves this complexity and an extension to image persistence. An implementation is available in Gudhi.

cs.CG↗
arXiv2026

Numerical Simulation of Transdermal Insulin Delivery Using a Coated Microneedle in a 2D Skin Model

In this work, we present a computational model to investigate transdermal insulin delivery using coated microneedles. A detailed skin geometry incorporating a coated microneedles was developed to analyze insulin release through the different skin layers and to evaluate the influence of key transport parameters. The model represents the major skin layers: the stratum corneum, viable epidermis, and dermis. Unstructured grids were used to achieve a reliable resolution of the model. The simulations provide insights into the permeation of insulin from the coated microneedles and the transport and distribution across the different skin layers. Finally, the simulation results were compared with experimental data to evaluate the predictive capability of the model.

cs.CG↗