{
  "id": "1202.2652",
  "version": "v2",
  "published": "2012-02-13T08:00:41.000Z",
  "updated": "2012-03-06T00:12:02.000Z",
  "title": "Ehrhart f*-coefficients of polytopal complexes are non-negative integers",
  "authors": [
    "Felix Breuer"
  ],
  "comment": "19 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart $\\delta$-vector), which is the vector of coefficients of $L_P$ with respect to a certain binomial basis and which coincides with the $h$-vector of a regular unimodular triangulation of $P$ (if one exists). One important result by Stanley about $h^*$-vectors of polytopes is that their entries are always non-negative. However, recent combinatorial applications of Ehrhart theory give rise to polytopal complexes with $h^*$-vectors that have negative entries. In this article we introduce the Ehrhart $f^*$-vector of polytopes or, more generally, of polytopal complexes $K$. These are again coefficient vectors of $L_K$ with respect to a certain binomial basis of the space of polynomials and they have the property that the $f^*$-vector of a unimodular simplicial complex coincides with its $f$-vector. The main result of this article is a counting interpretation for the $f^*$-coefficients which implies that $f^*$-coefficients of integral polytopal complexes are always non-negative integers. This holds even if the polytopal complex does not have a unimodular triangulation and if its $h^*$-vector does have negative entries. Our main technical tool is a new partition of the set of lattice points in a simplicial cone into discrete cones. Further results include a complete characterization of Ehrhart polynomials of integral partial polytopal complexes and a non-negativity theorem for the $f^*$-vectors of rational polytopal complexes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-03-06T00:12:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20",
      "52B70",
      "05A10",
      "05A15",
      "05E45",
      "11C08"
    ],
    "keywords": [
      "non-negative integers",
      "ehrhart polynomial",
      "integral partial polytopal complexes",
      "unimodular simplicial complex coincides",
      "coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.2652B"
    }
  }
}