{
  "id": "1607.07529",
  "version": "v1",
  "published": "2016-07-26T03:12:02.000Z",
  "updated": "2016-07-26T03:12:02.000Z",
  "title": "A bound for the index of a quadratic form after scalar extension to the function field of a quadric",
  "authors": [
    "Stephen Scully"
  ],
  "categories": [
    "math.NT",
    "math.AC",
    "math.AG"
  ],
  "abstract": "Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of a celebrated bound established in earlier work of Karpenko-Merkurjev and Totaro; on the other, it is a direct generalization of Karpenko's theorem on the possible values of the first higher isotropy index. We prove its validity in two important cases: (i) the case where $\\mathrm{char}(F) \\neq 2$, and (ii) the case where $\\mathrm{char}(F) = 2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic-geometric, and the second being purely algebraic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-26T03:12:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E04",
      "14E05",
      "15A03"
    ],
    "keywords": [
      "scalar extension",
      "function field",
      "first higher isotropy index",
      "anisotropic quadratic form",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}