Poincaré problem

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: 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.