{
  "id": "2202.10257",
  "version": "v2",
  "published": "2022-02-21T14:22:28.000Z",
  "updated": "2023-04-27T16:25:46.000Z",
  "title": "$S$-integral quadratic forms and homogeneous dynamics",
  "authors": [
    "Irving Calderón"
  ],
  "comment": "We add in the introduction a discussion of the extension of the results to any number field. Some misprints corrected",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $S = \\{ \\infty \\} \\cup S_f$ be a finite set of places of $\\mathbb{Q}$. Using homogeneous dynamics, we establish two new quantitative and explicit results about integral quadratic forms in three or more variables: The first is a criterion of $S$-integral equivalence. The second determines a finite generating set of any $S$-integral orthogonal group. Both theorems--which extend results of H. Li and G. Margulis for $S = \\{ \\infty\\}$--are given by polynomial bounds on the size of the coefficients of the quadratic forms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-04-27T16:25:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E12",
      "37A17",
      "11E08",
      "11H55",
      "37A25"
    ],
    "keywords": [
      "integral quadratic forms",
      "homogeneous dynamics",
      "integral orthogonal group",
      "theorems-which extend results",
      "explicit results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}