{
  "id": "1401.6667",
  "version": "v2",
  "published": "2014-01-26T16:50:00.000Z",
  "updated": "2014-01-31T16:46:50.000Z",
  "title": "Ranks of Quotients, Remainders and $p$-Adic Digits of Matrices",
  "authors": [
    "Mustafa Elsheikh",
    "Andy Novocin",
    "Mark Giesbrecht"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT",
    "cs.SC"
  ],
  "abstract": "For a prime $p$ and a matrix $A \\in \\mathbb{Z}^{n \\times n}$, write $A$ as $A = p (A \\,\\mathrm{quo}\\, p) + (A \\,\\mathrm{rem}\\, p)$ where the remainder and quotient operations are applied element-wise. Write the $p$-adic expansion of $A$ as $A = A^{[0]} + p A^{[1]} + p^2 A^{[2]} + \\cdots$ where each $A^{[i]} \\in \\mathbb{Z}^{n \\times n}$ has entries between $[0, p-1]$. Upper bounds are proven for the $\\mathbb{Z}$-ranks of $A \\,\\mathrm{rem}\\, p$, and $A \\,\\mathrm{quo}\\, p$. Also, upper bounds are proven for the $\\mathbb{Z}/p\\mathbb{Z}$-rank of $A^{[i]}$ for all $i \\ge 0$ when $p = 2$, and a conjecture is presented for odd primes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-31T16:46:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A03",
      "15B33",
      "15B36",
      "11C20"
    ],
    "keywords": [
      "adic digits",
      "upper bounds",
      "odd primes",
      "quotient operations",
      "adic expansion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.6667E"
    }
  }
}