{
  "abstract": "Let D_N be the determinant of the generic symmetric N x N matrix, and let differential operators act by ordinary differentiation over a field of odd characteristic p. We determine the apolar ideal of D_N in every size. In addition to the classical quadratic relations, it is generated by one (p-1) x (p-1) determinant of differential variables for each set of 2p-2 indices. These additional generators are minimal modulo the quadratic ideal. An exterior-algebra realization identifies the remaining apolar relations with radicals of invariant pairings for SL_2. Semisimplification gives a Hilbert series in terms of Dyck paths of height at most p-2, and the Steinberg symmetrizer gives the explicit generators of the entire radical. We also compute the ranks of powers of a general linear form. The weak Lefschetz property holds exactly when N <= 2p-2, and the strong Lefschetz property holds exactly when N < p.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38212",
  "author_total": 1,
  "authors": [
    {
      "name": "Ying Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ht6595ncktdx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38212",
  "fulltext_url": null,
  "id": "8ht6595ncktdx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38212",
  "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",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7215073259",
  "pdf_url": "https://arxiv.org/pdf/2609.38212",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ht6595ncktdx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38212",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38212"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Symmetric-determinant apolarity in odd characteristic",
  "title_text": "Symmetric-determinant apolarity in odd characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/8ht6595ncktdx",
  "venue": "arXiv",
  "versions": []
}