{
  "id": "2410.22507",
  "version": "v1",
  "published": "2024-10-29T19:57:10.000Z",
  "updated": "2024-10-29T19:57:10.000Z",
  "title": "Universality criterion sets for quadratic forms over number fields",
  "authors": [
    "Vitezslav Kala",
    "Jakub Krásenský",
    "Giuliano Romeo"
  ],
  "comment": "19 pages, comments are welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "In analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset $C$ of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of $C$, then it necessarily represents all totally positive integers, i.e., is universal. We use a novel characterization of minimal criterion sets to show that they always exist and are unique, and that they must contain certain explicit elements. We also extend the uniqueness result to the more general setting of representations of a given subset of the integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-29T19:57:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E12",
      "11E20",
      "11E25",
      "11R04",
      "11R80"
    ],
    "keywords": [
      "universality criterion sets",
      "totally positive integers",
      "totally real number field",
      "quadratic form represents",
      "minimal criterion sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}