{
  "id": "2003.02679",
  "version": "v1",
  "published": "2020-03-05T14:46:30.000Z",
  "updated": "2020-03-05T14:46:30.000Z",
  "title": "On the Ehrhart Polynomial of Minimal Matroids",
  "authors": [
    "Luis Ferroni"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "We provide a formula for the Ehrhart polynomial of the connected matroid of size $n$ and rank $k$ with the least number of bases, also known as a minimal matroid [9]. We prove that their polytopes are Ehrhart positive and $h^*$-real-rooted (and hence unimodal). We use our formula for these Ehrhart polynomials to prove that the operation of circuit-hyperplane relaxation of a matroid preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $h^*$-real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-05T14:46:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35",
      "52B20",
      "11B73"
    ],
    "keywords": [
      "ehrhart polynomial",
      "matroid preserves ehrhart positivity",
      "connected matroid",
      "corresponding uniform matroid",
      "corresponding minimal matroid"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}