{
  "id": "2406.15954",
  "version": "v1",
  "published": "2024-06-22T22:36:54.000Z",
  "updated": "2024-06-22T22:36:54.000Z",
  "title": "Hilbert's 13th problem in prime characteristic",
  "authors": [
    "Oakley Edens",
    "Zinovy Reichstein"
  ],
  "comment": "12 pages, 2 figures",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "The resolvent degree $\\textrm{rd}_{\\mathbb{C}}(n)$ is the smallest integer $d$ such that a root of the general polynomial $$f(x) = x^n + a_1 x^{n-1} + \\ldots + a_n$$ can be expressed as a composition of algebraic functions in at most $d$ variables with complex coefficients. It is known that $\\textrm{rd}_{\\mathbb{C}}(n) = 1$ when $n \\leqslant 5$. Hilbert was particularly interested in the next three cases: he asked if $\\textrm{rd}_{\\mathbb{C}}(6) = 2$ (Hilbert's Sextic Conjecture), $\\textrm{rd}_{\\mathbb{C}}(7) = 3$ (Hilbert's 13th Problem) and $\\textrm{rd}_{\\mathbb{C}}(8) = 4$ (Hilbert's Octic Conjecture). These problems remain open. It is known that $\\textrm{rd}_{\\mathbb{C}}(6) \\leqslant 2$, $\\textrm{rd}_{\\mathbb{C}}(7) \\leqslant 3$ and $\\textrm{rd}_{\\mathbb{C}}(8) \\leqslant 4$. It is not known whether or not $\\textrm{rd}_{\\mathbb{C}}(n)$ can be $> 1$ for any $n \\geqslant 6$. In this paper, we show that all three of Hilbert's conjectures can fail if we replace $\\mathbb C$ with a base field of positive characteristic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-22T22:36:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12E05",
      "14G17"
    ],
    "keywords": [
      "hilberts 13th problem",
      "prime characteristic",
      "hilberts sextic conjecture",
      "problems remain open",
      "hilberts octic conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}