Ax–Schanuel type theorems on functional transcendence via Nevanlinna theory

In this paper, we apply Nevanlinna theory to prove two Ax–Schanuel type theorems for functional transcendence when the original exponential map is replaced by other meromorphic functions. We give examples show that these results are optimal. As a byproduct, also analytic dependence implies algebraic certain classes of entire Finally, some links transcendental number and geometric theorem will be discussed.