{
  "id": "2112.15243",
  "version": "v2",
  "published": "2021-12-30T23:41:29.000Z",
  "updated": "2022-08-21T16:50:44.000Z",
  "title": "On quadratic Waring's problem in totally real number fields",
  "authors": [
    "Jakub Krásenský",
    "Pavlo Yatsyna"
  ],
  "comment": "16 pages; accepted in Proc. Am. Math. Soc",
  "categories": [
    "math.NT"
  ],
  "abstract": "We improve the bound of the $g$-invariant of the ring of integers of a totally real number field, where the $g$-invariant $g(r)$ is the smallest number of squares of linear forms in $r$ variables that is required to represent all the quadratic forms of rank $r$ that are representable by the sum of squares. Specifically, we prove that the $g_{\\mathcal{O}_K}(r)$ of the ring of integers $\\mathcal{O}_K$ of a totally real number field $K$ is at most $g_{\\mathbb{Z}}([K:\\mathbb{Q}]r)$. Moreover, it can also be bounded by $g_{\\mathcal{O}_F}([K:F]r+1)$ for any subfield $F$ of $K$. This yields a sub-exponential upper bound for $g(r)$ of each ring of integers (even if the class number is not $1$). Further, we obtain a more general inequality for the lattice version $G(r)$ of the invariant and apply it to determine the value of $G(2)$ for all but one real quadratic field.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-08-21T16:50:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E12",
      "11D85",
      "11E25",
      "11E39"
    ],
    "keywords": [
      "totally real number field",
      "quadratic warings problem",
      "real quadratic field",
      "sub-exponential upper bound",
      "smallest number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}