Ax-Schanuel type inequalities for functional transcendence via Nevanlinna theory

The Ax-Schanuel Theorem implies that for any $\mathbb{Q}$-linearly independent modulo $\mathbb{C}$ entire functions of one complex variable $f_1,...,f_n$, the transcendence degree over $\mathbb{C}$ of $f_1, ..., f_n, e(f_1),..., e(f_n)$ is at least $n+1$ where $e(z)=e^{2\pi i z}$. It is natural to ask what happens if one replaces the exponential map $e$ by some other meromorphic functions. In this talk, we will apply Nevanlinna theory to obtain several inequalities of the transcendence degree over $\mathbb{C}$ of $f_1, ..., f_n, F(f_1),..., F(f_n)$ when $f_i$'s are entire functions with some growth restrictions and $F$ is a transcendental meromorphic function. The results are joint work with Jiaxing Huang.