{
  "id": "2204.13202",
  "version": "v1",
  "published": "2022-04-27T21:26:07.000Z",
  "updated": "2022-04-27T21:26:07.000Z",
  "title": "Hilbert's 13th Problem for Algebraic Groups",
  "authors": [
    "Zinovy Reichstein"
  ],
  "comment": "40 pages",
  "categories": [
    "math.GR",
    "math.AC",
    "math.AG"
  ],
  "abstract": "The algebraic form of Hilbert's 13th Problem asks for the resolvent degree $\\text{rd}(n)$ of the general polynomial $f(x) = x^n + a_1 x^{n-1} + \\ldots + a_n$ of degree $n$, where $a_1, \\ldots, a_n$ are independent variables. The resolvent degree is the minimal integer $d$ such that every root of $f(x)$ can be obtained in a finite number of steps, starting with $\\mathbb C(a_1, \\ldots, a_n)$ and adjoining algebraic functions in $\\leq d$ variables at each step. Recently Farb and Wolfson defined the resolvent degree $\\text{rd}_k(G)$ of any finite group $G$ and any base field $k$ of characteristic $0$. In this setting $\\text{rd}(n) = \\text{rd}_{\\mathbb C}(S_n)$, where $S_n$ denotes the symmetric group. In this paper we define $\\text{rd}_k(G)$ for every algebraic group $G$ over an arbitrary field $k$, investigate the dependency of this quantity on $k$ and show that $\\text{rd}_k(G) \\leq 5$ for any field $k$ and any connected group $G$. The question of whether $\\text{rd}_k(G)$ can be bigger than $1$ for any field $k$ and any algebraic group $G$ over $k$ (not necessarily connected) remains open.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-27T21:26:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G10",
      "20G15"
    ],
    "keywords": [
      "algebraic group",
      "resolvent degree",
      "hilberts 13th problem asks",
      "remains open",
      "minimal integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}