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Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions

Authors: C. Ward Henson and Lee A. Rubel
Journal: Trans. Amer. Math. Soc. 282 (1984), 1-32
MSC: Primary 03B25; Secondary 03C05, 32A22
Correction: Trans. Amer. Math. Soc. 294 (1986), 381.
MathSciNet review: 728700
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Abstract: 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.

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