{
  "id": "1305.6242",
  "version": "v1",
  "published": "2013-05-27T14:44:30.000Z",
  "updated": "2013-05-27T14:44:30.000Z",
  "title": "Rational solutions of certain Diophantine equations involving norms",
  "authors": [
    "Maciej Ulas"
  ],
  "comment": "submitted",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this note we present some results concerning the unirationality of the algebraic variety $\\cal{S}_{f}$ given by the equation \\begin{equation*} N_{K/k}(X_{1}+\\alpha X_{2}+\\alpha^2 X_{3})=f(t), \\end{equation*} where $k$ is a number field, $K=k(\\alpha)$, $\\alpha$ is a root of an irreducible polynomial $h(x)=x^3+ax+b\\in k[x]$ and $f\\in k[t]$. We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\\in k\\setminus k^{3}$. We prove that if $\\op{deg}f=4$ and the variety $\\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\\neq 0$, then $\\cal{S}_{f}$ is $k$-unirational. A similar result is proved for a broad family of quintic polynomials $f$ satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of $\\cal{S}_{f}$ (with non-trivial $k$-rational point) is proved for any polynomial $f$ of degree 6 with $f$ not equivalent to the polynomial $h$ satisfying the condition $h(t)\\neq h(\\zeta_{3}t)$, where $\\zeta_{3}$ is the primitive third root of unity. We are able to prove the same result for an extension of degree 3 generated by the root of polynomial $h(x)=x^3+ax+b\\in k[x]$, provided that $f(t)=t^6+a_{4}t^4+a_{1}t+a_{0}\\in k[t]$ with $a_{1}a_{4}\\neq 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-27T14:44:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D57",
      "11D85"
    ],
    "keywords": [
      "diophantine equations",
      "rational solutions",
      "rational point",
      "pure cubic extensions",
      "irreducible polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6242U"
    }
  }
}