Submission date: 30 May 2011.

Abstract:

We consider the theory of algebraically closed fields of characteristic zero with raising to powers operations. In an earlier paper we have described complete first-order theories of such structures, provided that a diophantine conjecture CIT does hold. Here we get rid of this assumption. The theory of complex numbers with raising to real powers satisfies the description if Schanuel's conjecture holds. In particular, we have proved that a (weaker) version of Schanuel's conjecture implies that every well-defined system of exponential sums with real exponents has a solution. Recent result by Bays, Kirby and Wilkie states that the required version of Schanuel's conjecture holds for almost every choice of exponents. It follows that for the corresponding choice of real exponents we have an unconditional description of the first order theory of the complex numbers with raising to these powers.

Mathematics Subject Classification: 03C98, 11J81

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