{
  "id": "2309.10866",
  "version": "v1",
  "published": "2023-09-19T18:26:35.000Z",
  "updated": "2023-09-19T18:26:35.000Z",
  "title": "Determination of hyperovals by lines through a few points",
  "authors": [
    "Zhiguo Ding",
    "Michael E. Zieve"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "If S is a set of q+2 points in P^2(F_q) such that some point of S is not on any line containing two other points of S, then in suitable coordinates S has the form S_f:={(c:f(c):1) : c in F_q} U {(1:0:0),(0:1:0)} for some f(X) in F_q[X]. Let T be a subset of S_f which contains the two infinite points and at least 3+log_3(q)/4 finite points. We show that if there is no line passing through a point of T and two other points of S_f, and deg(f)<=q^{1/4}, then no three points of S_f are collinear, so that S_f is a hyperoval. We also determine all f(X) with deg(f)<=q^{1/4} for which S_f is a hyperoval, which strengthens a result that was proved by Caullery and Schmidt using entirely different methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-19T18:26:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B25",
      "11T06",
      "51E20"
    ],
    "keywords": [
      "determination",
      "infinite points",
      "suitable coordinates",
      "strengthens"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}