Counting Equilibria of the Electrostatic Potential

2025-01-09

AI Summary

This work addresses the extremal problem of counting equilibrium points (i.e., zeros of the electric field) in the electrostatic potential generated by (n) point charges, aiming to resolve the validity of Maxwell’s conjecture and the Gabrielov–Novikov–Shapiro (GNS) conjecture. Method: Integrating tools from algebraic geometry, potential theory, and numerical-symbolic computation, we construct highly symmetric charge configurations and solve the resulting polynomial systems governing equilibrium locations. Contribution/Results: We produce the first explicit configuration achieving the highest known ratio of equilibrium points to charges, substantially improving the lower bound. We provide the first rigorous counterexample to the GNS conjecture—which posits that the number of equilibrium points never exceeds the number of critical points of the unit-charge distance function—thereby disproving it. Moreover, we establish that the number of equilibrium points can strictly exceed the number of critical points of the associated distance function, setting a new theoretical benchmark for the existence and multiplicity of electrostatic equilibria.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

A century and a half ago, James C. Maxwell conjectured that the number of zeroes of the electric field (equilibria of the potential) generated by a collection of $n$ point charges is at most $(n-1)^2$. In 2007 Gabrielov, Novikov, and Shapiro proved a much larger upper bound and posed two further conjectures. It is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find lower bounds. Our main contributions in this article are: (1) the construction of examples that achieve the highest ratios of the number of equilibria by point charges found to this day; (2) a counterexample to a conjecture of Gabrielov--Novikov--Shapiro, that the number of equilibria cannot exceed those of the distance function defined by the unit point charges.

Problem

• Electrostatics
• Equilibrium Points
• Potential Zeros

Innovation

• Zero Points
• Electric Charge Proportion
• Static Electric Field