{
  "id": "1404.5866",
  "version": "v1",
  "published": "2014-04-23T15:33:51.000Z",
  "updated": "2014-04-23T15:33:51.000Z",
  "title": "On the Density of Integer Points on the Generalised Markoff-Hurwitz and Dwork Hypersurfaces",
  "authors": [
    "Igor E. Shparlinski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We use bounds of mixed character sums modulo a prime $p$ to estimate the density of integer points on the hypersurface $$ f_1(x_1) + \\ldots + f_n(x_n) =a x_1^{k_1} \\ldots x_n^{k_n} $$ for some polynomials $f_i \\in {\\mathbb Z}[X]$, nonzero integer $a$ and positive integers $k_i$ $i=1, \\ldots, n$. In the case of $$ f_1(X) = \\ldots = f_n(X) = X^2 \\quad \\text{and}\\quad k_1 = \\ldots = k_n =1 $$ the above congruence is known as the Markoff-Hurwitz hypersurface, while for $$ f_1(X) = \\ldots = f_n(X) = X^n\\quad \\text{and}\\quad k_1 = \\ldots = k_n =1 $$ it is known as the Dwork hypersurface. Our result is substantially stronger than those known for general hypersurfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-23T15:33:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integer points",
      "dwork hypersurface",
      "generalised markoff-hurwitz",
      "mixed character sums modulo",
      "general hypersurfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.5866S"
    }
  }
}