Schneider-Lang theorem ($|S|\le [K:\mathbb{Q}]({\rho }_1+{\rho }_2)$)

Let $f_1,\dots ,f_s$ be meromorphic functions with $f_j^{\prime }\in K[f_1,\dots ,f_s]$ for a fixed number field $K$. Suppose $f_1,f_2$ are algebraically independent over $\mathbb{C}$ and have finite orders ${\rho }_1,{\rho }_2$. The set $S$ of points where every $f_j$ is finite and belongs to $K$ satisfies the displayed bound. The orders are growth orders for meromorphic functions. For $f_1=z$, $f_2=e^z$, the orders are zero and one and the functions are algebraically independent; an algebraic value of $e^{\alpha }$ at nonzero algebraic $\alpha$ would produce infinitely many such points $n\alpha$.