{
  "id": "2403.11943",
  "version": "v2",
  "published": "2024-03-18T16:37:53.000Z",
  "updated": "2024-11-22T11:54:37.000Z",
  "title": "Galois groups of random polynomials over the rational function field",
  "authors": [
    "Alexei Entin"
  ],
  "comment": "v2: added some references",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a fixed prime power $q$ and natural number $d$ we consider a random polynomial $$f=x^n+a_{n-1}(t)x^{n-1}+\\ldots+a_1(t)x+a_0(t)\\in\\mathbb F_q[t][x]$$ with $a_i$ drawn uniformly and independently at random from the set of all polynomials in $\\mathbb F_q[t]$ of degree $\\le d$. We show that with probability tending to 1 as $n\\to\\infty$ the Galois group $G_f$ of $f$ over $\\mathbb F_q(t)$ is isomorphic to $S_{n-k}\\times C$, where $C$ is cyclic, $k$ and $|C|$ are small quantities with a simple explicit dependence on $f$. As a corollary we deduce that $\\mathbb P(G_f=S_n\\,|\\,f\\mbox{ irreducible})\\to 1$ as $n\\to\\infty$. Thus we are able to overcome the $S_n$ versus $A_n$ ambiguity in the most natural small box random polynomial model over $\\mathbb F_q[t]$, which has not been achieved over $\\mathbb Z$ so far.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-11-22T11:54:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational function field",
      "galois group",
      "small box random polynomial model",
      "natural small box random polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}