{
  "id": "math/0308050",
  "version": "v3",
  "published": "2003-08-06T08:54:26.000Z",
  "updated": "2008-12-17T15:27:45.000Z",
  "title": "Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors",
  "authors": [
    "Thomas Voigt",
    "Günter M. Ziegler"
  ],
  "comment": "9 pages",
  "journal": "Combinatorics, Probability & Computing, 15:463-471, 2006",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "Let $P(d)$ be the probability that a random 0/1-matrix of size $d \\times d$ is singular, and let $E(d)$ be the expected number of 0/1-vectors in the linear subspace spanned by d-1 random independent 0/1-vectors. (So $E(d)$ is the expected number of cube vertices on a random affine hyperplane spanned by vertices of the cube.) We prove that bounds on $P(d)$ are equivalent to bounds on $E(d)$: \\[ P(d) = (2^{-d} E(d) + \\frac{d^2}{2^{d+1}}) (1 + o(1)). \\] We also report about computational experiments pertaining to these numbers.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-12-17T15:27:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "05B20",
      "05D40"
    ],
    "keywords": [
      "hyperplanes",
      "expected number",
      "random independent",
      "random affine hyperplane",
      "cube vertices"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......8050V"
    }
  }
}