{
  "id": "math/0403546",
  "version": "v1",
  "published": "2004-03-31T16:39:04.000Z",
  "updated": "2004-03-31T16:39:04.000Z",
  "title": "Neighborhood complexes and generating functions for affine semigroups",
  "authors": [
    "Herbert E. Scarf",
    "Kevin M. Woods"
  ],
  "categories": [
    "math.CO",
    "math.OC"
  ],
  "abstract": "Given a_1,a_2,...,a_n in Z^d, we examine the set, G, of all non-negative integer combinations of these a_i. In particular, we examine the generating function f(z)=\\sum_{b\\in G} z^b. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in Z^n. In the generic case, this follows from algebraic results of D. Bayer and B. Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-03-31T16:39:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "90C10"
    ],
    "keywords": [
      "generating function",
      "neighborhood complexes",
      "affine semigroups",
      "maximal lattice-free bodies",
      "arbitrary lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......3546S"
    }
  }
}