{
  "id": "1901.06893",
  "version": "v1",
  "published": "2019-01-21T11:55:50.000Z",
  "updated": "2019-01-21T11:55:50.000Z",
  "title": "Minimal tropical basis for Bergman fan of matroid",
  "authors": [
    "Yasuhito Nakajima"
  ],
  "comment": "10 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and $R_{10}$. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-21T11:55:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35"
    ],
    "keywords": [
      "bergman fan",
      "unique minimal tropical basis",
      "simple binary matroid",
      "simple regular matroid",
      "cographic matroids"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}