{
  "abstract": "I give a short proof that the border determinantal complexity of $F_n=x_1^n+\\cdots+x_n^n$ over $\\mathbb{C}$ is at least $(n-1)^2/(4e)$, and at least $(n-1)^2/(2e)$ in the symmetric model. The same argument shows that the border determinantal complexity of $x_1^d+\\cdots+x_N^d$ is $Θ(Nd)$ for all $N\\ge3$ and $d\\ge2$. The earlier article arXiv:2606.13628, by the same author, introduced the first superlinear lower bounds and includes verification by Lean formalization. However, the AI written exposition was hard to follow. Kumar and Volk (arXiv:2609.34462) give a short proof of the exact ordinary case, but do not address the border or symmetric cases. In this article, I provide a greatly simplified exposition of the border and symmetric cases, using the implicit function theorem and an explicit section of simple solutions in place of the conormal specialization arguments of arXiv:2606.13628. I choose an explicit polynomial system with $n(n-1)^{n-2}$ simple solutions for $F_n$. I then use the implicit function theorem to ensure that each point persists under small perturbations. Finally, kernel equations and the multihomogeneous Bézout theorem bound the number of these points for a small determinant. The exposition is due to the human author. The mathematical simplification was obtained in discussion with ChatGPT Astra and refined in conversation with Claude Opus 5.5.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.11090",
  "author_total": 1,
  "authors": [
    {
      "name": "Karthik Sheshadri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6zce8gxnxkg3s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.11090",
  "fulltext_url": null,
  "id": "6zce8gxnxkg3s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.11090",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7164184610",
  "pdf_url": "https://arxiv.org/pdf/2606.11090",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6zce8gxnxkg3s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2606.11090",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.11090"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A short proof of a quadratic lower bound on the border determinantal complexity of $x_1^n+\\cdots+x_n^n$ over $\\mathbb{C}$: A simple human exposition",
  "title_text": "A short proof of a quadratic lower bound on the border determinantal complexity of x_1^n+⋯+x_n^n over ℂ: A simple human exposition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/6zce8gxnxkg3s",
  "venue": "arXiv",
  "versions": []
}