{
  "abstract": "The Elekes-Szabó theorem establishes that an irreducible algebraic hypersurface $Z(F)$ contains few grid points unless it exhibits a specific group-related structure. Identifying this structure from the polynomial $F$ is a challenging problem in combinatorial geometry. Our first main result (Theorem 2.3) provides a local geometric criterion to detect such group-related hypersurfaces. By applying this criterion, we develop a geometric framework for boundary varieties, which are defined by the vanishing of partial derivatives along $Z(F)$. In Theorem 2.4, we show that for group-related varieties, these boundary varieties must be contained in coordinate slices. This gives a strict geometric constraint on the loci where $Z(F)$ becomes tangent to coordinate directions. As an application, we study configurations formed by $d$ coordinate-grid hyperplane families together with a one-parameter polynomial family of hyperspheres in $\\mathbb{R}^d$. If one chooses $n$ members from each of these $d+1$ families and obtains $Ω(n^{d-η})$ common incidence points, then the hypersphere family is forced to have a very restricted form: it is concentric in dimensions $d \\geq 3$, and in dimension $2$ it is either concentric or consists of fixed-radius circles whose centres lie on a line parallel to a coordinate axis. We also generalize the pinned distance problem initiated by Elekes and Szabó for three points in the plane to $d+1$ points in $\\mathbb{R}^d$. More precisely, in Theorem 2.8 we prove that if $d+1$ families of hyperspheres centred at fixed points determine $Ω(n^{d-η})$ points, each lying on one hypersphere from each family, then the centres must be affinely dependent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.03366",
  "author_total": 1,
  "authors": [
    {
      "name": "Mehdi Makhul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e7cxm3ehkcd0r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.03366",
  "fulltext_url": null,
  "id": "e7cxm3ehkcd0r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.03366",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7167582139",
  "pdf_url": "https://arxiv.org/pdf/2607.03366",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e7cxm3ehkcd0r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2607.03366",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.03366"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Geometric Criterion for Degeneracy in the Elekes-Szabó Theorem",
  "title_text": "A Geometric Criterion for Degeneracy in the Elekes-Szabó Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/e7cxm3ehkcd0r",
  "venue": "arXiv",
  "versions": []
}