{
  "abstract": "Let $σ_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. Previous work determined $C(σ_{n,d})$ when $d$ is odd or a power of two, but the general even non-power-of-two case remained open. We determine the certificate complexity for every degree $1\\le d\\le n$, thereby completing the classification for elementary symmetric Boolean functions. Writing $d=2^t m$ with $m$ odd, we obtain an explicit formula in which the possible deficit from the maximal value $n$ is controlled by $2^t$, while the exact value is determined by a binary containment condition involving $m$. In particular, \\[ n-2^{ν_2(d)}+1\\le C(σ_{n,d})\\le n, \\] and we characterize when the upper bound is attained. Moreover, we determine the least positive period of the certificate-complexity deficit $Δ_d(n)=n-C(σ_{n,d})$: it is $1$ for odd $d$, equals $d$ when $d$ is a power of two, and equals $2^{\\lfloor\\log_2 d\\rfloor+1}$ for even non-power-of-two $d$. This least-period problem is distinct from the classical periodicity of the underlying value sequence $\\binom{j}{d}\\bmod 2$. The known odd-degree and power-of-two formulas are recovered as special cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15678",
  "author_total": 2,
  "authors": [
    {
      "name": "Jing Zhang"
    },
    {
      "name": "Yuan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e2xhy6g5q59v6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15678",
  "fulltext_url": null,
  "id": "e2xhy6g5q59v6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15678",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7213334514",
  "pdf_url": "https://arxiv.org/pdf/2609.15678",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e2xhy6g5q59v6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.15678",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15678"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Certificate Complexity of Elementary Symmetric Boolean Functions of Arbitrary Degree",
  "title_text": "Certificate Complexity of Elementary Symmetric Boolean Functions of Arbitrary Degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/e2xhy6g5q59v6",
  "venue": "arXiv",
  "versions": []
}