Schanuel’s Conjecture: algebraic independence of transcendental numbers

Abstract

Schanuel’s conjecture asserts that given linearly independent complex numbers x ₁, …, x _n , there are at least n algebraically independent numbers among the 2n numbers

$$ {x_1}, \ldots ,{x_n},{\rm{ }}{e^{{x_1}}}, \ldots ,{e^{{x_n}}}.$$

This simple statement has many remarkable consequences; we explain some of them. We also present the state of the art on this topic.