{
  "id": "2411.14960",
  "version": "v1",
  "published": "2024-11-22T14:18:10.000Z",
  "updated": "2024-11-22T14:18:10.000Z",
  "title": "First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability",
  "authors": [
    "Alexandra Shlapentokh",
    "Caleb Springer"
  ],
  "comment": "40 pages",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "In this paper, we study questions of definability and decidability for infinite algebraic extensions ${\\bf K}$ of $\\mathbb{F}_p(t)$ and their subrings of $\\mathcal{S}$-integral functions. We focus on fields ${\\bf K}$ satisfying a local property which we call $q$-boundedness. This can be considered a function field analogue of prior work of the first author which considered algebraic extensions of $\\mathbb{Q}$. One simple consequence of our work states that if ${\\bf K}$ is a $q$-bounded Galois extension of $\\mathbb{F}_p(t)$, then the integral closure $\\mathcal{O}_{\\bf K}$ of $\\mathbb{F}_p[t]$ inside ${\\bf K}$ is first-order definable in ${\\bf K}$. Under the additional assumption that the constant subfield of ${\\bf K}$ is infinite, it follows that both $\\mathcal{O}_{\\bf K}$ and ${\\bf K}$ have undecidable first-order theories. Our primary tools are norm equations and the Hasse Norm Principle, in the spirit of Rumely. Our paper has an intersection with a recent arXiv preprint by Mart\\'inez-Ranero, Salcedo, and Utreras, although our definability results are more extensive and undecidability results are much stronger.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-22T14:18:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "12L05",
      "11U09"
    ],
    "keywords": [
      "integral functions",
      "first-order definitions",
      "undecidability",
      "hasse norm principle",
      "function field analogue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}