{
  "id": "1508.00830",
  "version": "v1",
  "published": "2015-08-04T16:52:43.000Z",
  "updated": "2015-08-04T16:52:43.000Z",
  "title": "Height bounds on zeros of quadratic forms over $\\overline{\\mathbb Q}$",
  "authors": [
    "Lenny Fukshansky"
  ],
  "comment": "23 pages; to appear in Research in Mathematical Sciences. arXiv admin note: substantial text overlap with arXiv:1307.0564",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we establish three results on small-height zeros of quadratic polynomials over $\\overline{\\mathbb Q}$. For a single quadratic form in $N \\geq 2$ variables on a subspace of $\\overline{\\mathbb Q}^N$, we prove an upper bound on the height of a smallest nontrivial zero outside of an algebraic set under the assumption that such a zero exists. For a system of $k$ quadratic forms on an $L$-dimensional subspace of $\\overline{\\mathbb Q}^N$, $N \\geq L \\geq \\frac{k(k+1)}{2}+1$, we prove existence of a nontrivial simultaneous small-height zero. For a system of one or two inhomogeneous quadratic and $m$ linear polynomials in $N \\geq m+4$ variables, we obtain upper bounds on the height of a smallest simultaneous zero, if such a zero exists. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations and contributes to the literature of so-called \"absolute\" Diophantine results with respect to height. All bounds on height are explicit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-04T16:52:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G50",
      "11E12",
      "11E39"
    ],
    "keywords": [
      "height bounds",
      "upper bound",
      "smallest nontrivial zero outside",
      "single quadratic form",
      "nontrivial simultaneous small-height zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150800830F"
    }
  }
}