{
  "id": "1906.12054",
  "version": "v1",
  "published": "2019-06-28T06:24:08.000Z",
  "updated": "2019-06-28T06:24:08.000Z",
  "title": "On the functional graph generated by a plane curve and its quadratic twist",
  "authors": [
    "Bernard Mans",
    "Min Sha",
    "Jeffrey Smith",
    "Daniel Sutantyo"
  ],
  "comment": "32 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Given a plane curve defined by $Y^2 = f(X)$ over a finite field $\\mathbb{F}_q$ of odd characteristic and a non-square element $\\lambda$ in $\\mathbb{F}_q$, we define a functional graph by choosing the elements in $\\mathbb{F}_q$ as vertices and drawing an edge from $x$ to $y$ if and only if $(x,y)$ is either a point on $Y^2 = f(X)$ or a point on $\\lambda Y^2 = f(X)$. We show that if $f$ is a permutation polynomial over $\\mathbb{F}_q$, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials $f$ of low degree, it turns out that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-28T06:24:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "plane curve",
      "functional graph",
      "quadratic twist",
      "hamiltonian cycle",
      "permutation polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}