{
  "abstract": "We determine the sharp total-support uncertainty for the standard complete set of mutually unbiased bases in every prime dimension at least five, proving that the minimum is \\(T_s(p)=p^2-p+2\\). This resolves all prime dimensions left open by Fiorentino and Weigert, beginning with dimension eleven, and gives a uniform analytic determination of the optimal lower bound in dimension seven, where the earlier determination was partly numerical. The main new ingredient is a sharp zero-count theorem for arbitrary complex Fourier data supported on the finite parabola. A cyclotomic reduction and truncated local expansion produce a first nonzero finite-field jet that is affine in the transverse variable, reducing the planar zero count to a one-dimensional multiplicity problem. We then classify all extremizing projective states and derive an exact enumeration formula. In dimension seven, this gives analytic four-point extremizers and exactly \\(12348\\) such rays, completing the analytic description and count left open in the earlier work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16030",
  "author_total": 1,
  "authors": [
    {
      "name": "Dongwei Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cn5w0jkcp901f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16030",
  "fulltext_url": null,
  "id": "cn5w0jkcp901f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16030",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213408197",
  "pdf_url": "https://arxiv.org/pdf/2609.16030",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cn5w0jkcp901f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.16030",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16030"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sharp Total-Support Uncertainty for the Standard Complete Set of Mutually Unbiased Bases",
  "title_text": "Sharp Total-Support Uncertainty for the Standard Complete Set of Mutually Unbiased Bases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/cn5w0jkcp901f",
  "venue": "arXiv",
  "versions": []
}