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

Abstract

In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x plus y"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x+y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, multiplication <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x dot y"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mi>y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \cdot y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and two-place exponentiation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript y"> <mml:semantics> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>y</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">x^y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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.