{
  "id": "1403.7787",
  "version": "v2",
  "published": "2014-03-30T17:20:27.000Z",
  "updated": "2014-10-20T07:11:50.000Z",
  "title": "Integer decomposition property of free sums of convex polytopes",
  "authors": [
    "Takayuki Hibi",
    "Akihiro Higashitani"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Let $\\mathcal{P} \\subset \\mathbb{R}^{d}$ and $\\mathcal{Q} \\subset \\mathbb{R}^e$ be integral convex polytopes of dimension $d$ and $e$ which contain the origin of $\\mathbb{R}^{d}$ and $\\mathbb{R}^e$, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of $\\mathcal{P}$ and $\\mathcal{Q}$ to possess the integer decomposition property will be presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-30T17:20:27.000Z",
      "title": "Facets of $(0, 1)$-polytopes with squarefree initial ideals",
      "abstract": "Let $\\mathcal{P} \\subset \\mathbb{R}^{d}$ be a $(0, 1)$-polytope of dimension $d$ which contains the origin of $\\mathbb{R}^{d}$. Suppose that the configuration $\\mathcal{A}$ arising from $\\mathcal{P}$ satisfies $\\mathbb{Z} \\mathcal{A} = \\mathbb{Z}^{d+1}$ and that the toric ideal $I_{\\mathcal{A}}$ of $\\mathcal{A}$ possesses a squarefree initial ideal with respect to the reverse lexicographic order induced by an ordering of the variables for which the variable corresponding to the origin is the weakest. It is then proved that the equation of each facet of $\\mathcal{P}$ is of the form $\\sum_{i=1}^{d} a_{i}z_{i} = b$, where each $a_{i}$ is an integer and where $b \\in \\{0, 1\\}$.",
      "comment": "6 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-20T07:11:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20",
      "13P10"
    ],
    "keywords": [
      "squarefree initial ideal",
      "toric ideal",
      "reverse lexicographic order",
      "configuration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}