Eremenko's conjecture on higher-order Briot--Bouquet equations

The Briot--Bouquet theorem describes the meromorphic solutions of autonomous first-order algebraic differential equations. We prove Eremenko's conjecture extending it to all orders: if $f$ is meromorphic in $\mathbb{C}$ and $P(f,f^{(n)})=0$ for a nonzero polynomial $P$, then $f$ is rational, ellipti