{
  "id": "2102.07465",
  "version": "v1",
  "published": "2021-02-15T11:21:19.000Z",
  "updated": "2021-02-15T11:21:19.000Z",
  "title": "On parametric and generic polynomials with one parameter",
  "authors": [
    "Pierre Dèbes",
    "Joachim König",
    "François Legrand",
    "Danny Neftin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given fields $k \\subseteq L$, our results concern one parameter $L$-parametric polynomials over $k$, and their relation to generic polynomials. The former are polynomials $P(T,Y) \\in k[T][Y]$ of group $G$ which parametrize all Galois extensions of $L$ of group $G$ via specialization of $T$ in $L$, and the latter are those which are $L$-parametric for every field $L \\supseteq k$. We show, for example, that being $L$-parametric with $L$ taken to be the single field $\\mathbb{C}((V))(U)$ is in fact sufficient for a polynomial $P(T, Y) \\in \\mathbb{C}[T][Y]$ to be generic. As a corollary, we obtain a complete list of one parameter generic polynomials over a given field of characteristic 0, complementing the classical literature on the topic. Our approach also applies to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer conjecture, we provide one parameter families of affine curves over number fields, all with a rational point, but with no rational generic point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-15T11:21:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational generic point",
      "parameter generic polynomials",
      "galois extensions",
      "rational point",
      "number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}