{
  "id": "1603.01349",
  "version": "v1",
  "published": "2016-03-04T05:27:07.000Z",
  "updated": "2016-03-04T05:27:07.000Z",
  "title": "Two arithmetic relations of a subspace of $P^{n}(K)$ and a subgroup of $\\operatorname{Aut}(K)$",
  "authors": [
    "Jun-ichi Matsushita"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a commutative field and let $V$ be a subspace of $P^{n}(K)$. Let $\\Gamma$ be a subgroup of $\\operatorname{Aut}(K)$ and define the action of $\\Gamma$ on $P^{n}(K)$ by letting $\\sigma((x_{i})_{0\\leqslant i\\leqslant n})$ be $(\\sigma(x_{i}))_{0\\leqslant i\\leqslant n}$ for $\\sigma\\in\\Gamma$, $(x_{i})_{0\\leqslant i\\leqslant n}\\in P^{n}(K)$. In this paper, using two arithmetic notions defined in terms of the Pl\\\"{u}cker coordinates of $V$ and the invariant field of $\\Gamma$, we answer the questions: By what number is the dimension of the join (resp. meet) of $\\sigma(V)$ ($\\sigma\\in\\Gamma$) greater (resp. less) than the dimension of $V$?",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-04T05:27:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic relations",
      "arithmetic notions",
      "invariant field",
      "commutative field",
      "coordinates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160301349M"
    }
  }
}