{
  "id": "1711.06021",
  "version": "v1",
  "published": "2017-11-16T10:56:21.000Z",
  "updated": "2017-11-16T10:56:21.000Z",
  "title": "Probabilities of incidence between lines and a plane curve over finite fields",
  "authors": [
    "Mehdi Makhul",
    "Josef Schicho",
    "Matteo Gallet"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points. In particular, we focus on the limits of these probabilities under successive finite field extensions. The main tools we use are the Lang-Weil bound for the number of rational points of an algebraic variety and a geometric interpretation of the Galois group of a curve. Supposing absolute irreducibility for the curve, we prove the existence of these limits, and under a mildly stronger condition we provide an explicit formula for them, depending only on the degree of the curve.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-16T10:56:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "plane curve",
      "probability",
      "successive finite field extensions",
      "lang-weil bound",
      "stronger condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}