{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]] of dimension $n$, and let $\\mathcal{M}(X)$ be its field of [[meromorphic functions|W6FTSF57T1]].\nThen the [[transcendence degree|D7TR0YWR9H]] of $\\mathcal{M}(X)$ over $\\C$ is at most $n$.\nEquivalently, any $n + 1$ meromorphic functions on $X$ satisfy a nonzero polynomial relation with complex coefficients.\n\n## Remarks\n\nProof idea: let $f_1, \\ldots, f_{n+1}$ be meromorphic on $X$ and consider all polynomials of degree at most $k$ in them.\nThese form a space of dimension of order $k^{n+1}$.\nCover $X$ by finitely many polydiscs and use a Schwarz lemma.\nThis shows that such a polynomial expression vanishing to high order at finitely many chosen points vanishes identically.\nThe number of conditions needed grows only like $k^n$.\nFor large $k$ some nonzero polynomial in the $f_i$ is therefore identically zero.\n\nExample: on a compact Riemann surface ($n = 1$), any two meromorphic functions are algebraically dependent; for $\\hat\\C$, $f$ and $g$ are both rational functions of $z$.\n\nCompactness is necessary: on the non-compact manifold $\\C$, the meromorphic functions $z$ and $e^z$ are algebraically independent, so the transcendence degree is at least $2 \u003e 1$.\n\nIn terms of the [[algebraic dimension|F0GF0JEZTG]], the theorem says $a(X) \\le \\dim X$.\nThe field $\\mathcal{M}(X)$ is moreover finitely generated over $\\C$, so it is the function field of an algebraic variety of dimension $a(X)$.\n\n## Explanations\n\n### L1 human\n\nOn a compact complex manifold of dimension $n$ there cannot be more than $n$ independent meromorphic functions.\nAny $n + 1$ of them are tied together by a polynomial equation, just as $x$ and $y$ are tied by $x^2 + y^2 = 1$ on a circle.\nThis mirrors algebraic geometry, where an $n$-dimensional variety has a function field of transcendence degree $n$.\n\n### L2 human\n\nThe theorem makes the [[algebraic dimension|F0GF0JEZTG]] a number between $0$ and $\\dim X$, and the case of equality defines [[Moishezon manifolds|8ZW9P9JRQD]].\nIt shows that compact complex manifolds behave like algebraic varieties as far as their meromorphic functions go, even when they are far from algebraic.\nThe proof uses only local estimates and compactness, in the spirit of the Schwarz lemma.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "F0GF0JEZTG"
    ],
    "kind": "statement",
    "msc": [
      "32J10",
      "32A20"
    ],
    "name": "Siegel's theorem on meromorphic function fields",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Moishezon_manifold"
      }
    ],
    "tag": "ENXCWPTG92",
    "uses": [
      "JSGWAN2RJV",
      "W6FTSF57T1",
      "D7TR0YWR9H"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/ENXCWPTG92/",
  "links": [
    {
      "field": "about",
      "target": "F0GF0JEZTG"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    },
    {
      "field": "uses",
      "target": "W6FTSF57T1"
    },
    {
      "field": "uses",
      "target": "D7TR0YWR9H"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ENXCWPTG92.md",
  "msc": [
    "32J10",
    "32A20"
  ],
  "name": "Siegel's theorem on meromorphic function fields",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ENX/ENXCWPTG92.md",
  "status": null,
  "tag": "ENXCWPTG92",
  "uri": "m2://ENXCWPTG92",
  "url": "https://tamnd.github.io/math2/wiki/siegels-theorem-on-meromorphic-function-fields/",
  "writing": {
    "tag": "ENXCWPTG92",
    "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
      }
    ]
  }
}