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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:

[1] Annals of Pure and Applied Logic 132 pp 67– (2004)
[2] DOI: 10.1090/S0894-0347-96-00216-0 · Zbl 0892.03013
[3] Stable theories with a new predicate 66 pp 1127– (2001) · Zbl 1002.03023
[4] A shorter model theory (1997) · Zbl 0873.03036
[5] A remark on Zilber’s pseudoexponentiation 71 pp 791– (2006) · Zbl 1112.03029
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