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**Some applications of Nevanlinna theory to mathematical logic: Identities of exponential functions.**
*(English)*
Zbl 0533.03015

Summary: In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition \(x+y\), multiplication \(x\cdot y\), and two-place exponentiation \(x^ y\). For a restricted class of such functions, we show that every true identity follows from the natural set of eleven axioms. The rates of growth of such functions, in the case of a single independent variable x, as \(x\to \infty\), are also studied, and we give an algorithm for the Hardy relation of eventual domination, again for a restricted class of functions. Value distribution of analytic functions of one and of several complex variables, especially the Nevanlinna characteristic, plays a major role in our proofs.

### MSC:

03C05 | Equational classes, universal algebra in model theory |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |

32A22 | Nevanlinna theory; growth estimates; other inequalities of several complex variables |

03B25 | Decidability of theories and sets of sentences |