{
  "id": "2112.11839",
  "version": "v2",
  "published": "2021-12-22T12:30:16.000Z",
  "updated": "2023-05-17T22:53:53.000Z",
  "title": "Two Formulas for $F$-Polynomials",
  "authors": [
    "Feiyang Lin",
    "Gregg Musiker",
    "Tomoki Nakanishi"
  ],
  "journal": "International Mathematics Research Notices (April 2023)",
  "doi": "10.1093/imrn/rnad074",
  "categories": [
    "math.CO"
  ],
  "abstract": "We discuss a product formula for $F$-polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for $F$-polynomials. The other is based on the Fock-Goncharov decomposition of mutations. We conclude by expanding this product formula as a sum and illustrate applications. This expansion provides an explicit combinatorial computation of $F$-polynomials in a given seed that depends only on the $\\mathbf{c}$-vectors and $\\mathbf{g}$-vectors along a finite sequence of mutations from the initial seed to the given seed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-05-17T22:53:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E16"
    ],
    "keywords": [
      "polynomials",
      "product formula",
      "explicit combinatorial computation",
      "finite sequence",
      "mutation rule"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}