Ax-Schanuel Type Theorems on Functional Transcendence via Nevanlinna Theory

TL;DR

This paper uses Nevanlinna Theory to establish Ax-Schanuel type theorems for various meromorphic functions, linking analytic and algebraic dependence, and exploring implications for transcendental number theory and geometry.

Contribution

It extends Ax-Schanuel theorems to broader classes of meromorphic functions beyond the exponential, revealing new connections between analytic and algebraic dependence.

Findings

Proves Ax-Schanuel type theorems for meromorphic functions using Nevanlinna Theory

Shows analytic dependence implies algebraic dependence for certain entire functions

Discusses implications for transcendental number theory and geometric Ax-Schanuel

Abstract

We will apply Nevanlinna Theory to prove several Ax-Schanuel type Theorems for functional transcendence when the exponential map is replaced by other meromorphic functions. We also show that analytic dependence will imply algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax-Schanuel Theorem will be discussed.