{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nLet $X$ be a connected [[complex manifold|JSGWAN2RJV]] and let $f$ be a [[meromorphic function|W6FTSF57T1]] on $X$.\nFind [[holomorphic functions|507C34W0ZT]] $g$ and $h$ on $X$, with $h$ not identically zero, such that $f = g / h$ on $X$.\nAlso ask that the germs of $g$ and $h$ be coprime in the local ring $\\mathcal{O}_{X,x}$ at every point $x \\in X$.\n\n## Remarks\n\nWithout the coprimality condition this is the weak Poincaré problem.\nHenri Poincaré proved in 1883 that every meromorphic function on $\\C^2$ is a quotient of two entire functions.\nLocally, every meromorphic function is a quotient of coprime holomorphic germs, because the ring $\\mathcal{O}_{X,x}$ is a unique factorization domain.\nThe local denominators form data for the [[second Cousin problem|D44KAF8PNK]]: a global solution $h$ of it gives $g = f h$ holomorphic, and $f = g / h$ with coprime germs.\nSo the Poincaré problem is solvable on every manifold where all second Cousin problems are solvable, for example on a Stein manifold $X$ with $H^2(X, \\Z) = 0$, such as $\\C^n$.\nOn every Stein manifold the weak Poincaré problem is solvable.\n\n## Explanations\n\n### L1 human\n\nIn one variable every meromorphic function on the plane is a quotient of two entire functions, by the Weierstrass factorization theorem.\nThe Poincaré problem asks the same in several variables, with the extra wish that numerator and denominator share no common factor at any point.\nIt reduces to the second Cousin problem, the problem of building a global function with prescribed zeros, and it has a positive answer on $\\C^n$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "W6FTSF57T1"
    ],
    "history": {
      "first_stated": "1883",
      "posed_by": "Henri Poincaré"
    },
    "kind": "problem",
    "msc": [
      "32A20",
      "32E10"
    ],
    "name": "Poincaré problem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cousin_problems"
      }
    ],
    "tag": "R1W62VM8W6",
    "uses": [
      "JSGWAN2RJV",
      "W6FTSF57T1",
      "507C34W0ZT",
      "D44KAF8PNK"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/R1W62VM8W6/",
  "links": [
    {
      "field": "about",
      "target": "W6FTSF57T1"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    },
    {
      "field": "uses",
      "target": "W6FTSF57T1"
    },
    {
      "field": "uses",
      "target": "507C34W0ZT"
    },
    {
      "field": "uses",
      "target": "D44KAF8PNK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/R1W62VM8W6.md",
  "msc": [
    "32A20",
    "32E10"
  ],
  "name": "Poincaré problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R1W/R1W62VM8W6.md",
  "status": null,
  "tag": "R1W62VM8W6",
  "uri": "m2://R1W62VM8W6",
  "url": "https://tamnd.github.io/math2/wiki/poincare-problem/",
  "writing": {
    "tag": "R1W62VM8W6",
    "kind": "problem",
    "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
      }
    ]
  }
}