Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

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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.