{
  "id": "0902.1072",
  "version": "v3",
  "published": "2009-02-06T11:52:44.000Z",
  "updated": "2009-11-26T09:06:49.000Z",
  "title": "Combinatorics and Genus of Tropical Intersections and Ehrhart Theory",
  "authors": [
    "Reinhard Steffens",
    "Thorsten Theobald"
  ],
  "comment": "Small revisions",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work who considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary $k$ and arbitrary Newton polytopes $P_1, ..., P_k$. This provides new formulas for the number of faces and the genus in terms of mixed volumes. By establishing some aspects of a mixed version of Ehrhart theory we show that the genus of a tropical intersection curve equals the genus of a toric intersection curve corresponding to the same Newton polytopes.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-11-26T09:06:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B70",
      "52C45",
      "14M25",
      "52B20"
    ],
    "keywords": [
      "ehrhart theory",
      "combinatorics",
      "tropical intersection curve equals",
      "arbitrary newton polytopes",
      "study combinatorial questions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.1072S"
    }
  }
}