{
  "id": "math/0703722",
  "version": "v2",
  "published": "2007-03-24T17:10:53.000Z",
  "updated": "2007-09-13T19:54:53.000Z",
  "title": "Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in R(x, y)",
  "authors": [
    "Valéry Mahé"
  ],
  "comment": "63 pages, a proposition has been added (proposition 2.8)",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions. As explained by Huisman and Mahe, a given monic squarefree positive polynomial in two variables x and y of degree in y divisible by 4 is a sum of three squares of rational fractions if and only if the jabobian variety of some hyperelliptic curve (associated to P) has an \"antineutral\" point. Using this criterium, we follow a method developped by Cassels, Ellison and Pfister to solve our problem : at first we show the Mordell-Weil rank of the jacobian variety J associated to some polynomial is zero (this step is done by doing a 2-descent), and then we check that the jacobian variety J has no antineutral torsion point.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-09-13T19:54:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H40",
      "14G05",
      "14H05",
      "14P99",
      "14Q05"
    ],
    "keywords": [
      "hyperelliptic curve",
      "rational fractions",
      "jacobian variety",
      "hilberts seventeenth problem",
      "antineutral torsion point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3722M"
    }
  }
}