{
  "id": "math/0702406",
  "version": "v1",
  "published": "2007-02-14T09:59:35.000Z",
  "updated": "2007-02-14T09:59:35.000Z",
  "title": "Explicit Formula for Counting Lattice Points of Polyhedra",
  "authors": [
    "Jean B. Lasserre",
    "Eduardo S. Zeron"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Given $z\\in C^n$ and $A\\in Z^{m\\times n}$, we consider the problem of evaluating the counting function $h(y;z):=\\sum\\{z^x : x\\in Z^n; Ax=y, x\\geq 0\\}$. We provide an explicit expression for $h(y;z)$ as well as an algorithm with possibly numerous but very simple calculations. In addition, we exhibit finitely many fixed convex cones, explicitly and exclusively defined by $A$, such that for any $y\\in Z^m$, the sum $h(y;z)$ can be obtained by a simple formula involving the evaluation of $\\sum z^x$ over the integral points of those cones only. At last, we also provide an alternative (and different) formula from a decomposition of the generating function into simpler rational fractions, easy to invert.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-02-14T09:59:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "51M20",
      "90C57"
    ],
    "keywords": [
      "counting lattice points",
      "explicit formula",
      "simpler rational fractions",
      "simple calculations",
      "fixed convex cones"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......2406L"
    }
  }
}