Ax-Schanuel type inequalities for functional transcendence via Nevan-linna theory

Abstract

The Ax-Schanuel Theorem implies that for any Q-linearly independent modulo C entire functions of one complex variable f1,...,fn, the transcendence degree over C of f1,...,fn,e(f1),...,e(fn) is at least n+1 where e(z)=e2πiz. 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 C of f1,...,fn,F(f1),...,F(fn) when fi's are entire functions with some growth restrictions and F is a transcendental meromorphic function. The results are joint work with Jiaxing Huang.