{
  "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 $\\mathbf a_i\\in\\mathbf{R}^3$, we show that the Coulomb field has at most $2^{N-4}(N-1)(9N^2+9N+10)$ equilibria in the region where $S(\\mathbf x):=\\sum_iq_i|\\mathbf x-\\mathbf a_i|^{-3}$ 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 $S$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03965",
  "author_total": 2,
  "authors": [
    {
      "name": "Alberto Enciso"
    },
    {
      "name": "Daniel Peralta-Salas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7hptvfext45dd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03965",
  "fulltext_url": null,
  "id": "7hptvfext45dd",
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  "landing_url": "https://arxiv.org/abs/2609.03965",
  "language": null,
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  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
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      "arxiv",
      "math.CA"
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      "2608|Advanced Differential Equations and Dynamical Systems"
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  "openalex": "W7208774878",
  "pdf_url": "https://arxiv.org/pdf/2609.03965",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7hptvfext45dd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:31Z",
      "native_id": "2609.03965",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03965"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The finiteness conjecture for equilibria of electric fields generated by point charges of one sign",
  "title_text": "The finiteness conjecture for equilibria of electric fields generated by point charges of one sign",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/7hptvfext45dd",
  "venue": "arXiv",
  "versions": []
}