{
  "id": "math/0602593",
  "version": "v2",
  "published": "2006-02-26T18:11:37.000Z",
  "updated": "2006-03-16T19:00:59.000Z",
  "title": "Lattice polytopes with a given $h^*$-polynomial",
  "authors": [
    "Victor Batyrev"
  ],
  "comment": "10 pages. AMS-LaTeX, some typos were corrected",
  "categories": [
    "math.CO",
    "math.AC",
    "math.AG"
  ],
  "abstract": "Let $\\Delta \\subset \\R^n$ be an $n$-dimensional lattice polytope. It is well-known that $h_{\\Delta}^*(t) := (1-t)^{n+1} \\sum_{k \\geq 0} |k\\Delta \\cap \\Z^n| t^k $ is a polynomial of degree $d \\leq n$ with nonnegative integral coefficients. Let $AGL(n, \\Z)$ be the group of invertible affine integral transformations which naturally acts on $\\R^n$. For a given polynomial $h^* \\in \\Z[t]$, we denote by $C_{h^*}(n)$ the number $AGL(n, \\Z)$-equivalence classes of $n$-dimensional lattice polytopes such that $h^* = h_{\\Delta}^*(t)$. In this paper we show that $\\{C_{h^*}(n) \\}_{n \\geq 1}$ is a monotone increasing sequence which eventually becomes constant. This statement follows from a more general combinatorial result whose proof uses methods of commutative algebra.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-03-16T19:00:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20",
      "13H10",
      "14M25"
    ],
    "keywords": [
      "dimensional lattice polytope",
      "polynomial",
      "general combinatorial result",
      "invertible affine integral transformations",
      "equivalence classes"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......2593B"
    }
  }
}