{
  "abstract": "Let $S\\subset\\R^n_+$ be a compact convex set with $0\\in S$, and let $\\cP^S$ be the ring of polynomials whose exponents of the $m$-th degree lie in the dilates $mS$. Sigurðardóttir proved an Oka--Weil theorem for compact $\\cP^S$-convex subsets of $(\\C^*)^n$ and asked whether it holds for all compact $\\cP^S$-convex subsets of $\\C^n$. We answer this question affirmatively when the cone $\\R_+S$ is rational polyhedral, for every $S$ when $n=2$, and for several classes of nonpolyhedral cones in all dimensions; in all these cases the compact sets may meet the coordinate hyperplanes. For arbitrary $S$ we prove that the Oka--Weil property of a compact $\\cP^S$-convex set is equivalent to the compactness of its hull with respect to a finitely generated monomial subring of $\\cP^S$, to a corona-type certificate, and to the existence of a bounded neighbourhood defined by finitely many inequalities $|p_j|<c_j$ with $p_j\\in\\cP^S$. The proofs use monomial charts, the classical Oka--Weil theorem in an auxiliary affine space, and elementary Gelfand theory. As applications we obtain an Oka--Stolzenberg criterion for $\\cP^S$-hulls and the corresponding rational approximation theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21181",
  "author_total": 2,
  "authors": [
    {
      "name": "Duong Thi Thuy Van"
    },
    {
      "name": "Kieu Phuong Chi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/25c2v01e025m4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21181",
  "fulltext_url": null,
  "id": "25c2v01e025m4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21181",
  "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.CV"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7222554021",
  "pdf_url": "https://arxiv.org/pdf/2610.21181",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/25c2v01e025m4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21181",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21181"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Oka-Weil approximation theorem for polynomials with exponents in compact convex sets in $\\mathbb R^n$",
  "title_text": "Oka-Weil approximation theorem for polynomials with exponents in compact convex sets in ℝ^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/25c2v01e025m4",
  "venue": "arXiv",
  "versions": []
}