---
tag: R1W62VM8W6
kind: problem
name: Poincaré problem
msc: [32A20, "32E10"]
uses: [JSGWAN2RJV, W6FTSF57T1, 507C34W0ZT, D44KAF8PNK]
about: [W6FTSF57T1]
sources:
  - url: https://en.wikipedia.org/wiki/Cousin_problems
history:
  posed_by: "Henri Poincaré"
  first_stated: "1883"
---

## Problem

Let $X$ be a connected [[complex manifold|JSGWAN2RJV]] and let $f$ be a [[meromorphic function|W6FTSF57T1]] on $X$.
Find [[holomorphic functions|507C34W0ZT]] $g$ and $h$ on $X$, with $h$ not identically zero, such that $f = g / h$ on $X$.
Also ask that the germs of $g$ and $h$ be coprime in the local ring $\mathcal{O}_{X,x}$ at every point $x \in X$.

## Remarks

Without the coprimality condition this is the weak Poincaré problem.
Henri Poincaré proved in 1883 that every meromorphic function on $\C^2$ is a quotient of two entire functions.
Locally, every meromorphic function is a quotient of coprime holomorphic germs, because the ring $\mathcal{O}_{X,x}$ is a unique factorization domain.
The 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.
So 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 $\C^n$.
On every Stein manifold the weak Poincaré problem is solvable.

## Explanations

### L1 human

In one variable every meromorphic function on the plane is a quotient of two entire functions, by the Weierstrass factorization theorem.
The Poincaré problem asks the same in several variables, with the extra wish that numerator and denominator share no common factor at any point.
It reduces to the second Cousin problem, the problem of building a global function with prescribed zeros, and it has a positive answer on $\C^n$.