{
  "abstract": "For integers $N\\ge0$ and $n\\ge2$, let \\[ Δ_N^{(n)} =\\left\\{α\\in\\mathbb Z_{\\ge 0}^n: α_1+\\cdots+α_n=N\\right\\}. \\] We study discrete unique continuation for functions on this lattice simplex. For an integer $R\\ge1$, let $g:Δ_{nR}^{(n)}\\to\\mathbb R$ satisfy \\[ \\sum_{i=1}^n g(β+e_i)=0, \\qquad β\\inΔ_{nR-1}^{(n)}, \\] where $e_i$ is the $i$th standard basis vector. We prove that nonvanishing at the center implies \\[ |\\operatorname{supp} g|\\ge c_n R^{\\lceil n/2\\rceil}. \\] Here $\\operatorname{supp} g$ is the set of points where $g$ is nonzero, and $c_n>0$ depends only on $n$. The exponent $\\lceil n/2\\rceil$ is optimal. The proof represents the values of $g$ as polynomial coefficients and uses a Pascal uncertainty principle, which bounds from below the total number of nonzero coefficients of a one-variable polynomial and its unit translate in terms of their degree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02707",
  "author_total": 1,
  "authors": [
    {
      "name": "Linjun Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7mgme61m56j8c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02707",
  "fulltext_url": null,
  "id": "7mgme61m56j8c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02707",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7172527583",
  "pdf_url": "https://arxiv.org/pdf/2608.02707",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7mgme61m56j8c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2608.02707",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02707"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Discrete Unique Continuation on Simplex",
  "title_text": "Discrete Unique Continuation on Simplex",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/7mgme61m56j8c",
  "venue": "arXiv",
  "versions": []
}