The Transcendental Algebraic Geometry of Exponential and Power Maps

This paper explores the intricate interplay between transcendental number theory and algebraic geometry through the lens of exponential and power maps. We delve into the "transcendental algebraic geometry" that emerges when considering the algebraic relations between values of $e^z$ and $z^alpha$ for algebraic $z$ and $alpha$. The fundamental objective is to understand the geometric structures underlying sets defined by exponential and power functions, particularly focusing on their intersection with algebraic varieties. We investigate the extent to which these transcendental functions exhibit algebraic behavior, drawing upon concepts from model theory, Diophantine approximation, and classical transcendental number theory, such as Baker's theory of linear forms in logarithms and Schanuel's conjecture. The paper outlines a theoretical framework for analyzing the transcendence properties of points on algebraic varieties under the action of these non-algebraic maps, contributing to the broader program of characterizing algebraically closed fields with an exponentiation.