Schanuel’s Conjecture and Algebraic Roots of Exponential Polynomials

In this paper, I will prove that assuming Schanuel’s conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any exponential polynomial. This implies a special case of Shapiro’s conjecture: if p(x) and q(x) are two exponential polynomials over the algebraic numbers, each involving only one iteration of the exponential map, and they have common factors only of the form exp(g) for some exponential polynomial g, then p and q have only finitely many common zeros.