From 12 to 6: Sharpening the Three-Charge Bound in Maxwell's Problem

A. Gabrielov, Dm. Novikov, T. Novikov, B. Shapiro

Abstract

In \cite{GNS} we proved that, for every $α>0$, the potential of three positive point charges has at most $12$ nondegenerate equilibrium points. We also observed that the same method would give the sharper bound $6$ if a certain auxiliary polynomial system $Q=R=0$ had at least four solutions, counted with multiplicity, in each open quadrant of the $(f,g)$-plane. Here we prove this four-solution statement. The main new ingredient is a separation argument at the unique saddle point of a separated-variable first integral. Consequently, the upper bound for three charges improves from $12$ to $6$.

Disclosure

“tions of point charges has resurfaced anew. Acknowledgments Parts of this work, including the saddle-separation argument and the symbolic and numerical verification, were developed with the assistance of Claude (Anthropic). All results were independently verified by the authors. This project was mainly carried out while D. Novikov was visiting the Institute for Advanced Study. He thanks the Institute for its hospitality and excellent working c”
PDF page 9

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T19:14:40.233265Z

Structural counts

Pages 10 pdf

Theorems 1 source

Lemmas 4 source

Propositions 2 source

Corollaries 0 source

Definitions 0 source

Displayed equations 47 source

Bibliography entries 9 source

Appendix pages 10 estimated

Count notes

Source counts use the expanded primary TeX file gns-twelve-to-six-revised-MSC.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.