{
  "id": "2007.06454",
  "version": "v1",
  "published": "2020-07-13T15:39:16.000Z",
  "updated": "2020-07-13T15:39:16.000Z",
  "title": "Hermite reduction and a Waring's problem for integral quadratic forms over number fields",
  "authors": [
    "Wai Kiu Chan",
    "Maria Ines Icaza"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over $\\mathbb Q$ and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally real number field $K$. We apply the balanced HKZ-reduction theory to study the growth of the {\\em $g$-invariants} of the ring of integers of $K$. More precisely, for each positive integer $n$, let $\\mathcal O$ be the ring of integers of $K$ and $g_{\\mathcal O}(n)$ be the smallest integer such that every sum of squares of $n$-ary $\\mathcal O$-linear forms must be a sum of $g_{\\mathcal O}(n)$ squares of $n$-ary $\\mathcal O$-linear forms. We show that when $K$ has class number 1, the growth of $g_{\\mathcal O}(n)$ is at most an exponential of $\\sqrt{n}$. This extends the recent result obtained by Beli-Chan-Icaza-Liu on the growth of $g_{\\mathbb Z}(n)$ and gives the first sub-exponential upper bound for $g_{\\mathcal O}(n)$ for rings of integers $\\mathcal O$ other than $\\mathbb Z$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-13T15:39:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E12",
      "11E25",
      "11E39"
    ],
    "keywords": [
      "integral quadratic forms",
      "hermite reduction",
      "warings problem",
      "positive definite quadratic forms",
      "first sub-exponential upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}