The finiteness conjecture for equilibria of electric fields generated by point charges of one sign

Abstract

We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges q_i of possibly mixed signs at distinct sites a_i∈R³, we show that the Coulomb field has at most 2^N−4(N−1)(9N²+9N+10) equilibria in the region where S(x):=∑_iq_i|x−a_i|⁻³ does not vanish. Of course, for charges of one sign, S is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of S.

Quillen's finiteness conjecture for the cotangent complex in characteristic two
Timo Schürg
arXiv2 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples
Andres Ventas
arXiv15 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
On the bounded-conductor finiteness conjecture in equal characteristic
Yufan Luo, Yiqi Xu
arXiv10 Sep 2026PreprintCC BY-SA 4.0Full textAlgebra & number theoryarXiv
Arithmetic Constraints on Charges of Symmetric Monopoles
Paul Cusson
arXiv16 Oct 2026PreprintarXiv licenceFull textGeometry & topology Mathematical physicsarXiv
Homological finiteness properties of wreath products
Laurent Bartholdi, Yves De Cornulier, Dessislava H. Kochloukova
Quarterly Journal of Mathematics3 Nov 2026Peer-reviewedLicence not statedFull text12 citedAlgebra & number theoryHAL

Cite

BibTeX RIS

BibTeX

@misc{enciso2026finiteness,
  title = {{The finiteness conjecture for equilibria of electric fields generated by point charges of one sign}},
  author = {Alberto Enciso and Daniel Peralta-Salas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03965},
  eprint = {2609.03965},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03965},
}

RIS

TY  - UNPB
TI  - The finiteness conjecture for equilibria of electric fields generated by point charges of one sign
AU  - Alberto Enciso
AU  - Daniel Peralta-Salas
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03965
UR  - https://arxiv.org/abs/2609.03965
AB  - We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges q_i of possibly mixed signs at distinct sites a_i∈R^3, we show that the Coulomb field has at most 2^N−4(N−1)(9N^2+9N+10) equilibria in the region where S(x):=∑_iq_i|x−a_i|^−3 does not vanish. Of course, for charges of one sign, S is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of S.
ER  -

Abstract

Related

Cite