{
  "id": "1909.09537",
  "version": "v1",
  "published": "2019-09-20T14:42:23.000Z",
  "updated": "2019-09-20T14:42:23.000Z",
  "title": "A Note on the Decidability of $\\mbox{Th}_{\\exists}(\\mathbb{F}_q (t))$",
  "authors": [
    "Brian Tyrrell"
  ],
  "comment": "5 pages. Comments welcome",
  "categories": [
    "math.LO",
    "math.NT"
  ],
  "abstract": "This note proves that in the language of rings $\\mathcal{L}_{\\mbox{rings}} = \\{+, \\cdot, 0, 1\\}$ the existential theory of $\\mathbb{F}_q (t)$ is decidable, for any finite field $\\mathbb{F}_q$ of any characteristic. As a corollary we see $\\mbox{Th}_{\\exists}(\\mathbb{F}_q (t))$ is decidable in the language $\\mathcal{L}_{\\mbox{rings}} \\cup \\{F\\}$, where $F$ is a unary predicate expressing \"$x \\not \\in \\mathbb{F}_q$\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-20T14:42:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "decidability",
      "finite field",
      "existential theory",
      "characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}