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
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