{
  "id": "2310.09704",
  "version": "v1",
  "published": "2023-10-15T01:54:59.000Z",
  "updated": "2023-10-15T01:54:59.000Z",
  "title": "Explicit bounds for the solutions of superelliptic equations over number fields",
  "authors": [
    "Attila Bérczes",
    "Yann Bugeaud",
    "Kálmán Győry",
    "Jorge Mello",
    "Alina Ostafe",
    "Min Sha"
  ],
  "comment": "37 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a polynomial with coefficients in the ring $O_S$ of $S$-integers of a number field $K$, $b$ a non-zero $S$-integer, and $m$ an integer $\\ge 2$. We consider the equation $( \\star )$: $f(x) = b y^m$ in $x,y \\in O_S$. Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of $K, S, f, m$ and the $S$-norm of $b$ for the heights of the solutions $x$ of the equation $( \\star)$. Further, we give an explicit bound $C$ in terms of $K, S, f$ and the $S$-norm of $b$ such that if $m > C$ the equation $(\\star)$ has only solutions with $y = 0$ or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud, and extend earlier results of B\\'erczes, Evertse, and Gy\\H{o}ry to polynomials with multiple roots. In contrast with the previous results, our bounds depend on the $S$-norm of $b$ instead of its height.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-15T01:54:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "number field",
      "explicit bound",
      "superelliptic equations",
      "extend earlier results",
      "well-known leveque condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}