{
  "id": "math/0512132",
  "version": "v2",
  "published": "2005-12-06T18:24:05.000Z",
  "updated": "2007-03-21T20:33:46.000Z",
  "title": "Small zeros of quadratic forms over the algebraic closure of Q",
  "authors": [
    "Lenny Fukshansky"
  ],
  "comment": "17 pages; revised version per referee's request, in particular section 6 has been largely expanded; to appear in the International Journal of Number Theory",
  "journal": "International Journal of Number Theory, vol. 4 no. 3 (2008), pg. 503-523",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $N \\geq 2$ be an integer, $F$ a quadratic form in $N$ variables over $\\bar{\\mathbb Q}$, and $Z \\subseteq \\bar{\\mathbb Q}^N$ an $L$-dimensional subspace, $1 \\leq L \\leq N$. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space $(Z,F)$. This provides an analogue over $\\bar{\\mathbb Q}$ of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over $\\bar{\\mathbb Q}$. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over $\\bar{\\mathbb Q}$. This extends previous results of the author over number fields. All bounds on height are explicit.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-03-21T20:33:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E12",
      "11G50",
      "11H55",
      "11D09"
    ],
    "keywords": [
      "quadratic form",
      "algebraic closure",
      "small zeros",
      "bilinear space",
      "small-height maximal totally isotropic subspace"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12132F"
    }
  }
}