Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

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In this article, 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 such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, 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.

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Assuming Schanuel’s Conjecture we prove that for any irreducible variety V ⊆ ℂn × (ℂ*)n over ℚalg, of dimension n, and with dominant projections on both the first n coordinates and the last n coordinates, there exists a generic point \(\left( {\overline a ,{e^{\overline a }}} \right) \in V\). We obtain in this way many instances of the Strong Exponential Closure axiom introduced by Zilber in [20].
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