Generic points on exponential curves

TL;DR

Assuming Schanuel's conjecture, the paper proves that certain complex polynomials have infinitely many solutions of the form (z, e^z) that are algebraically independent.

Contribution

It establishes a conditional result on the existence of infinitely many algebraically independent solutions for a class of exponential polynomials.

Findings

Infinitely many algebraically independent solutions under Schanuel's conjecture

Results apply to irreducible polynomials with both variables present

Advances understanding of exponential algebraic independence

Abstract

We show, assuming Schanuel's conjecture, that every irreducible complex polynomial in two variables where both variables appear has infinitely many algebraically independent solutions of the form (z,e^z).