{
  "id": "0708.3619",
  "version": "v1",
  "published": "2007-08-27T14:41:45.000Z",
  "updated": "2007-08-27T14:41:45.000Z",
  "title": "Explicit Evaluation of Certain Exponential Sums of Quadratic Functions over $\\Bbb F_{p^n}$, $p$ Odd",
  "authors": [
    "Sandra Draper",
    "Xiang-dong Hou"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime and let $f(x)=\\sum_{i=1}^ka_ix^{p^{\\alpha_i}+1}\\in\\Bbb F_{p^n}[x]$, where $0\\le \\alpha_1<...<\\alpha_k$. We consider the exponential sum $S(f,n)=\\sum_{x\\in\\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\\pi i\\text{Tr}_n(y)/p}$, $y\\in\\Bbb F_{p^n}$, $\\text{Tr}_n=\\text{Tr}_{\\Bbb F_{p^n}/\\Bbb F_p}$. There is an effective way to compute the nullity of the quadratic form $\\text{Tr}_{mn}(f(x))$ for all integer $m>0$. Assuming that all such nullities are known, we find relative formulas for $S(f,mn)$ in terms of $S(f,n)$ when $\\nu_p(m) \\le \\min\\{\\nu_p(\\alpha_i):1\\le i\\le k\\}$, where $\\nu_p$ is the $p$-adic order. We also find an explicit formula for $S(f,n)$ when $\\nu_2(\\alpha_1)=...= \\nu_2(\\alpha_k)<\\nu_2(n)$. These results generalize those by Carlitz and by Baumert and McEliece. Parallel results with $p=2$ were obtained in a previous paper by the second author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-08-27T14:41:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T23"
    ],
    "keywords": [
      "exponential sum",
      "quadratic functions",
      "explicit evaluation",
      "adic order",
      "quadratic form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.3619D"
    }
  }
}