Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

TL;DR

Assuming Schanuel's conjecture, the paper proves that exponential polynomials over algebraic numbers have finitely many algebraic solutions, advancing understanding of exponential algebraic equations.

Contribution

It establishes finiteness of algebraic solutions for exponential polynomials assuming Schanuel's conjecture, connecting to Shapiro's conjecture and exponential field theory.

Findings

Finiteness of algebraic solutions under Schanuel's conjecture

Implication for Shapiro's conjecture in exponential fields

Applicability to pseudoexponential and algebraically closed exponential fields

Abstract

In this paper we prove that assuming Schanuel's conjecture, an exponential polynomial in one variable over the algebraic numbers has only finitely many algebraic solutions. This implies a positive answer to Shapiro's conjecture for exponential polynomials over the algebraic numbers for pseudoexponential fields as well as for any algebraically closed exponential field satisfying Schanuel's conjecture.