{
  "id": "math/0606708",
  "version": "v2",
  "published": "2006-06-28T08:50:43.000Z",
  "updated": "2006-12-04T16:03:35.000Z",
  "title": "On the unique representability of spikes over prime fields",
  "authors": [
    "Zhaoyang Wu",
    "Zhi-Wei Sun"
  ],
  "comment": "8 pages",
  "journal": "Discrete Math. 306(2006), 1798-1804",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "For an integer $n>2$, a rank-$n$ matroid is called an $n$-spike if it consists of $n$ three-point lines through a common point such that, for all $k\\in\\{1, 2, ..., n - 1\\}$, the union of every set of $k$ of these lines has rank $k+1$. Spikes are very special and important in matroid theory. In 2003 Wu found the exact numbers of $n$-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number $p$, a $GF(p$) representable $n$-spike $M$ is only representable on fields with characteristic $p$ provided that $n \\ge 2p-1$. Moreover, $M$ is uniquely representable over $GF(p)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-12-04T16:03:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35",
      "11B75",
      "11T99"
    ],
    "keywords": [
      "prime fields",
      "unique representability",
      "larger finite fields",
      "matroid theory",
      "exact numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6708W"
    }
  }
}