{
  "abstract": "Let $F$ be a field, let $0\\le \\ell\\le k-2$, and suppose that $n\\ge k-1$. We determine the minimum degree of a polynomial in $F[x_1,\\ldots,x_n]$ that vanishes to order at least $k$ at every nonzero vertex of the Boolean cube and to order exactly $\\ell$ at the origin. The answer is \\[ n+2k-2-ρ_F(k-\\ell), \\] where $ρ_F(s)$ is the least number of positive integers summing to $s-1$ whose corresponding Catalan numbers are nonzero in $F$. The proof gives an explicit basis of the reduced vanishing space. In this basis the top-degree map is diagonal, with Catalan numbers on the diagonal; a refinement using block coordinates shows that the basis is compatible with polynomial degree. In odd characteristic the degree is either $n+2k-3$ or $n+2k-4$, according as $C_{k-\\ell-2}$ is nonzero or zero. In characteristic $2$ it is \\[ n+2k-2-s_2(k-\\ell-1), \\] where $s_2$ denotes binary digit sum. Thus the first characteristic drop, and all later drops, are determined exactly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19009",
  "author_total": 1,
  "authors": [
    {
      "name": "Dean Menezes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bcnnc383413de.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19009",
  "fulltext_url": null,
  "id": "bcnnc383413de",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19009",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213523705",
  "pdf_url": "https://arxiv.org/pdf/2609.19009",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bcnnc383413de.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.19009",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19009"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Characteristic drops for high-order vanishing on the hypercube",
  "title_text": "Characteristic drops for high-order vanishing on the hypercube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/bcnnc383413de",
  "venue": "arXiv",
  "versions": []
}