{
  "id": "1411.7736",
  "version": "v1",
  "published": "2014-11-28T01:47:41.000Z",
  "updated": "2014-11-28T01:47:41.000Z",
  "title": "Local $h$-polynomials, invariants of subdivisions, and mixed Ehrhart theory",
  "authors": [
    "Eric Katz",
    "Alan Stapledon"
  ],
  "comment": "54 pages",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "There are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and $h^*$-polynomials, respectively. In this paper, we study their generalization to subdivisions and lattice subdivisions of polytopes. By abstracting constructions in mixed Hodge theory, we introduce multivariable polynomials which specialize to the $h$-, $h^*$- polynomials. These polynomials, the mixed $h$-polynomial and the (refined) limit mixed $h^*$-polynomial have rich symmetry, non-negativity, and unimodality properties, which both refine known properties of the classical polynomials, and reveal new structure. For example, we prove a lower bound theorem for a related invariant called the local $h^*$-polynomial. We introduce our polynomials by developing a very general formalism for studying subdivisions of Eulerian posets that extends the work of Stanley, Brenti and Athanasiadis on local $h$-vectors. In particular, we prove a conjecture of Nill and Schepers, and answer a question of Athanasiadis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-28T01:47:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20",
      "52B05",
      "14M25"
    ],
    "keywords": [
      "mixed ehrhart theory",
      "subdivisions",
      "lower bound theorem",
      "natural polynomial invariants",
      "lattice polytopes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.7736K"
    }
  }
}