{
  "abstract": "We count Boolean functions represented as the sign of a real polynomial of bounded degree on the $n$-dimensional cube. We prove that the logarithm of their number is asymptotic to that of the classical hyperplane-arrangement upper bound, uniformly over all degrees, with relative error $O((\\log n)^2/n)$. We also obtain sharper additive and relative error estimates in particular degree ranges, including an additive error $O(n^2)$ between the logarithms throughout a small linear degree range and improved bounds just below the central degree. The proof combines probabilistic estimates for Reed--Muller codes with a deletion argument for sets of evaluation vectors, an entropy count, and a sharp-threshold estimate for expected rank deficiency.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34378",
  "author_total": 1,
  "authors": [
    {
      "name": "Martin Anthony"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2891n87fwjffe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34378",
  "fulltext_url": null,
  "id": "2891n87fwjffe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34378",
  "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"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223821052",
  "pdf_url": "https://arxiv.org/pdf/2610.34378",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2891n87fwjffe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.34378",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34378"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Logarithmic asymptotics for polynomial threshold functions",
  "title_text": "Logarithmic asymptotics for polynomial threshold functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/2891n87fwjffe",
  "venue": "arXiv",
  "versions": []
}