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

Henson, C. Ward; Rubel, Lee A.

Trans. Am. Math. Soc. 282, 1-32 (1984).

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

Keywords:

identities between functions of many variables; rates of growth; algorithm for the Hardy relation of eventual domination; Value distribution of analytic functions; Nevanlinna characteristic