{
  "abstract": "The polynomial threshold degree (or threshold degree) of a Boolean function is the least degree of a real polynomial whose sign represents it. We prove that almost every Boolean function has polynomial threshold degree among the central one or two integers. More precisely, a uniformly chosen Boolean function on $n$ variables has threshold degree either $\\lfloor n/2\\rfloor$ or $\\lceil n/2\\rceil$, except with probability at most $\\exp(-c2^{n/2}/n)$, for an absolute constant $c>0$ and all sufficiently large $n$. The paper proves a 1991 conjecture of Aspnes, Beigel, Furst and Rudich. It also proves a 1991 conjecture of Wang and Williams for even $n$ and its main concentration assertion for odd $n$. A parity example gives a lower bound on the exceptional probability whose exponent differs from ours by a factor of order $n$. The proof relates polynomial sign representation to recovery of Reed--Muller codes from random erasures, and combines sharp-threshold estimates with bounds on generalized Hamming weights and minimum distance.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15057",
  "author_total": 1,
  "authors": [
    {
      "name": "Martin Anthony"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4zg9regmh43mq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15057",
  "fulltext_url": null,
  "id": "4zg9regmh43mq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15057",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
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      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
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  ],
  "openalex": "W7221807583",
  "pdf_url": "https://arxiv.org/pdf/2610.15057",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4zg9regmh43mq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.15057",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15057"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Almost every Boolean function has central threshold degree",
  "title_text": "Almost every Boolean function has central threshold degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/4zg9regmh43mq",
  "venue": "arXiv",
  "versions": []
}