{
  "id": "2302.06974",
  "version": "v1",
  "published": "2023-02-14T11:07:39.000Z",
  "updated": "2023-02-14T11:07:39.000Z",
  "title": "Quadratic equations in metabelian Baumslag-Solitar groups",
  "authors": [
    "Richard Mandel",
    "Alexander Ushakov"
  ],
  "comment": "17 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "For a finitely generated group $G$, the Diophantine problem over $G$ is the algorithmic problem of deciding whether a given equation $W(z_1,z_2,\\ldots,z_k) = 1$ (perhaps restricted to a fixed subclass of equations) has a solution in $G$. We investigate the algorithmic complexity of the Diophantine problem for the class $\\mathcal{C}$ of quadratic equations over the metabelian Baumslag-Solitar groups $BS(1,n)$. In particular, we prove that this problem is NP-complete whenever $n\\neq 1$, and determine the algorithmic complexity for various subclasses (orientable, nonorientable etc.) of $\\mathcal{C}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-14T11:07:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F10",
      "20F16",
      "68W30"
    ],
    "keywords": [
      "metabelian baumslag-solitar groups",
      "quadratic equations",
      "algorithmic complexity",
      "diophantine problem",
      "algorithmic problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}