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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].

In this paper we prove Shapiro's 1958 Conjecture on exponential polynomials,
assuming Schanuel's Conjecture.

We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

In this paper we define the category of exponential rings and develop some of its basic properties.

In 1984, Henson and Rubel [2] proved the following theorem: If p(x1,…, xn) is an exponential polynomial with coefficients in ℂ with no zeroes in ℂ, then p(x1,…, xn) = eg(x1,…, xn)
where g(x1,…, xn) is some exponential polynomial over C. In this paper, I will prove the analog of this theorem for Zilber's Pseudoexponential fields directly from the axioms. Furthermore, this proof relies only on the existential closedness axiom without any reference to Schanuel's conjecture.