{
  "id": "2207.03080",
  "version": "v1",
  "published": "2022-07-07T04:26:34.000Z",
  "updated": "2022-07-07T04:26:34.000Z",
  "title": "Diophantine equations of the form $Y^n=f(X)$ over function fields",
  "authors": [
    "Anwesh Ray"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\ell$ and $p$ be (not necessarily distinct) prime numbers and $F$ be a global function field of characteristic $\\ell$ with field of constants $\\kappa$. Let $f(X)$ be a polynomial in $X$ with coefficients in $\\kappa$. We study solutions to diophantine equations of the form $Y^{n}=f(X)$ which lie in $F$, and in particular, show that if $m$ and $f(X)$ satisfy additional conditions, then there are no non-constant solutions. The results obtained apply to the study of solutions to $Y^{n}=f(X)$ in certain $\\mathbb{Z}_{p}$-extensions of $F$ known as constant $\\mathbb{Z}_p$-extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \\dots, T_r]$, where $K$ is any field, showing that the only solutions must lie in $K$. We apply our methods to study solutions of diophantine equations of the form $Y^n=\\sum_{i=1}^d (X+ir)^m$, where $m,n, d\\geq 2$ are integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-07T04:26:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11R58",
      "11R23"
    ],
    "keywords": [
      "diophantine equations",
      "study solutions",
      "satisfy additional conditions",
      "global function field",
      "prime numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}