{
  "aliases": null,
  "backlinks": [
    {
      "field": "illustrates",
      "target": "27K5X7S4XN"
    }
  ],
  "body": "\n## Statement\n\nLet $(\\mathbf{x}, B)$ be a seed, with cluster $\\mathbf{x} = (x_1, \\dots, x_n)$ and a skew-symmetrizable integer matrix $B$, and let $\\mathcal{A}(B)$ be the coefficient-free [[cluster algebra|TKK0TV2MYC]] it generates.\nThen every cluster variable of $\\mathcal{A}(B)$ is a Laurent polynomial with integer coefficients in $x_1, \\dots, x_n$, that is, an element of the [[Laurent polynomial ring|QMY5JTZZB8]] $\\Z[x_1^{\\pm 1}, \\dots, x_n^{\\pm 1}]$.\n\n## Remarks\n\nThe theorem is due to Fomin and Zelevinsky.\nEach mutation divides by a cluster variable, so a priori the cluster variables are only rational functions of the initial ones.\nThe content of the theorem is that all these denominators are monomials.\nThe proof goes by induction on the length of a mutation sequence, and it uses that the exchange polynomials of neighbouring mutations are coprime.\nExample: for $n = 2$ and $B = \\begin{pmatrix} 0 \u0026 1 \\\\ -1 \u0026 0 \\end{pmatrix}$, the cluster variables satisfy $x_{m+1} x_{m-1} = x_m + 1$.\nStarting from $x_1, x_2$ one gets $x_3 = (x_2 + 1)/x_1$, $x_4 = (x_1 + x_2 + 1)/(x_1 x_2)$, $x_5 = (x_1 + 1)/x_2$, and then $x_6 = x_1$, $x_7 = x_2$.\nAll five are Laurent polynomials in $x_1, x_2$.\nThe special form of the exchange relations matters: a general rational recurrence produces denominators that are not monomials.\n\n## Explanations\n\n### L1 human\n\nSome recurrences divide at every step, yet the result never has a real denominator.\nThe Laurent phenomenon says this happens for all the variables of a cluster algebra: only powers of the starting variables ever appear in the denominators.\nSetting all starting variables to $1$ then gives integers, which explains why some nonlinear integer sequences stay integral.\n\n### L2 human\n\nThe Laurent phenomenon is one of the founding results of the theory of cluster algebras.\nSeveral nonlinear recurrences, such as the Somos-4 and Somos-5 recurrences, can be realized by mutations.\nSo their terms are Laurent polynomials in the initial values, which explains the integrality of the [[Somos sequences|27K5X7S4XN]] for $k = 4, 5$.\nA stronger statement, that the coefficients of these Laurent polynomials are nonnegative, was conjectured by Fomin and Zelevinsky and later proved for skew-symmetric matrices $B$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "TKK0TV2MYC"
    ],
    "kind": "statement",
    "msc": [
      "13F60",
      "39A20"
    ],
    "name": "Laurent phenomenon",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cluster_algebra"
      }
    ],
    "tag": "6YYPNKK6WZ",
    "uses": [
      "TKK0TV2MYC",
      "QMY5JTZZB8"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/6YYPNKK6WZ/",
  "links": [
    {
      "field": "about",
      "target": "TKK0TV2MYC"
    },
    {
      "field": "uses",
      "target": "TKK0TV2MYC"
    },
    {
      "field": "uses",
      "target": "QMY5JTZZB8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6YYPNKK6WZ.md",
  "msc": [
    "13F60",
    "39A20"
  ],
  "name": "Laurent phenomenon",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6YY/6YYPNKK6WZ.md",
  "status": null,
  "tag": "6YYPNKK6WZ",
  "uri": "m2://6YYPNKK6WZ",
  "url": "https://tamnd.github.io/math2/wiki/laurent-phenomenon/",
  "writing": {
    "tag": "6YYPNKK6WZ",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}