Sufficient conditions for a meromorphic function to be rational I know that rational functions are meromorphic, but under what conditions are meromorphic functions rational? I know that the automorphisms of the Riemann sphere are rational, but are there any more general conditions that ensure rationality?

The given function should have a finite number of poles on the Riemann sphere with the counting done with multiplicity given by the order of the pole. 
First consider the complex plane, i.e., the Riemann sphere without the point at infinity. Then by multiplying the meromorphic function, $f(z)$ with an entire function $q(z)=\prod_{a_i\in \text{poles}}(z-a_i)^{m_i}$ that vanishes with suitable multiplicity, $m_i$, at each pole, $a_i$, removes the  meromorphicity in the complex plane. You get an entire function, $p(z)=f(z) q(z)$ with no poles in the complex plane. The order of the pole at infinity is given to be finite. Thus, you need to argue that $p(z)$ is a polynomial in $z$. Then, you have shown that $f(z)=p(z)/q(z)$.
