{
  "abstract": "We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37180",
  "author_total": 2,
  "authors": [
    {
      "name": "Oliver Navío"
    },
    {
      "name": "Daniel Seco"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07c3wj5j7by87.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37180",
  "fulltext_url": null,
  "id": "07c3wj5j7by87",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37180",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
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  "openalex": "W7214918501",
  "pdf_url": "https://arxiv.org/pdf/2609.37180",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07c3wj5j7by87.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37180",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37180"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc",
  "title_text": "Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/07c3wj5j7by87",
  "venue": "arXiv",
  "versions": []
}