## On complex exponentiation restricted to the integers.(English)Zbl 1205.03051

The authors study model theory of expansions of algebraically closed fields $$\mathcal C$$ of characteristic 0 that are obtained by expanding the field by a ring $$Z({\mathcal C})$$ satisfying the first-order theory of the ring of integers and an exponential function EXP whose domain is restricted to the ring. In what follows, $${\mathbb C}$$ is the field of complex numbers, $${\mathbb Z}$$ is the ring of integers, $$\exp_{\epsilon}$$ is the exponential function $$x\mapsto \epsilon^x$$ restricted to $${\mathbb Z}$$, and exp is $$\exp_e$$. It is shown that if $$\epsilon$$ is algebraic, then $$\exp_\epsilon$$ is $$\emptyset$$-definable in $$({\mathbb C}, {\mathbb Z})$$, and if $$\epsilon$$ is transcendental then $$\exp_\epsilon$$ is not parametrically definable in $$({\mathbb C}, {\mathbb Z})$$. The authors give an axiomatization $$T$$ of the first-order theory of $$({\mathbb C}, {\mathbb Z},\exp)$$. The axioms of $$T$$ include the whole first-order theory of the ring $${\mathbb Z}$$ (the authors call it “the dark side of $$T$$”) and, among other axioms, the following Schanuel condition: For every positive integer $$k$$, if $$1=M_0, M_1,\dots, M_k\in Z({\mathcal C})$$ are linearly independent over the field of (standard) rational numbers, then $$\text{EXP}(M_0), \text{EXP}(M_1),\dots, \text{EXP}(M_k)$$ are algebraically independent over the field of fractions of $$Z({\mathcal C})$$. In the main result of the paper, it is shown that $$T$$ is complete, and hence it equals the first-order theory of $$({\mathbb C}, {\mathbb Z},\exp)$$.

### MSC:

 03C60 Model-theoretic algebra
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### References:

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