{
  "id": "0903.2508",
  "version": "v1",
  "published": "2009-03-13T22:19:57.000Z",
  "updated": "2009-03-13T22:19:57.000Z",
  "title": "Distribution of determinant of matrices with restricted entries over finite fields",
  "authors": [
    "Le Anh Vinh"
  ],
  "comment": "Journal of Combinatorics and Number Theory (to appear)",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "For a prime power $q$, we study the distribution of determinent of matrices with restricted entries over a finite field $\\mathbbm{F}_q$ of $q$ elements. More precisely, let $N_d (\\mathcal{A}; t)$ be the number of $d \\times d$ matrices with entries in $\\mathcal{A}$ having determinant $t$. We show that \\[ N_d (\\mathcal{A}; t) = (1 + o (1)) \\frac{|\\mathcal{A}|^{d^2}}{q}, \\] if $|\\mathcal{A}| = \\omega(q^{\\frac{d}{2d-1}})$, $d\\geqslant 4$. When $q$ is a prime and $\\mathcal{A}$ is a symmetric interval $[-H,H]$, we get the same result for $d\\geqslant 3$. This improves a result of Ahmadi and Shparlinski (2007).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-13T22:19:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11C20",
      "11T23"
    ],
    "keywords": [
      "finite field",
      "restricted entries",
      "distribution",
      "determinant",
      "prime power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.2508V"
    }
  }
}