{
  "abstract": "We determine the dimensions of all derivative loci and all higher multiplicity loci of an elementary symmetric polynomial over an algebraically closed field of positive characteristic. For the degree-d polynomial in n variables, set Q = p^{v_p(n-d+1)}. The common zero locus of its order-q derivatives has dimension d - q - 1 + 1_{Q>d-q}, whereas the locus of zeros of multiplicity at least s has dimension d - s + 1_{Q>d}. The order-two case settles the dimension question in a conjecture of Orzel, with a necessary correction when d is a power of p. We classify all components attaining the larger dimension and prove that they are generically smooth. Along the coordinate components in the smaller-dimension cases, we determine the local rings and obtain generic multiplicity exactly Q. Finally, we identify the full translation stabilizer scheme and derive a Frobenius normal form. These results distinguish dimension jumps from infinitesimal thickening and give a uniform description of their dependence on the characteristic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00259",
  "author_total": 1,
  "authors": [
    {
      "name": "Ying Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9bby4vf02ybs6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00259",
  "fulltext_url": null,
  "id": "9bby4vf02ybs6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00259",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7216418020",
  "pdf_url": "https://arxiv.org/pdf/2610.00259",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9bby4vf02ybs6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00259",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00259"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Derivative and multiplicity loci of elementary symmetric polynomials in positive characteristic",
  "title_text": "Derivative and multiplicity loci of elementary symmetric polynomials in positive characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/9bby4vf02ybs6",
  "venue": "arXiv",
  "versions": []
}