New Gelfond-Type Transcendental Numbers

TL;DR

This paper proves that solutions to certain exponential and algebraic equations involving transcendental functions are transcendental numbers, expanding understanding of their algebraic independence and properties.

Contribution

It establishes new results on the transcendence of solutions to equations involving exponential and polynomial functions, including exponential-polynomial equations.

Findings

Zeros of f(x)=g(x) with f as a transcendental function and g as a polynomial are transcendental.

Zeros of e^{f(x)}=g(x) with polynomial f and g are transcendental.

Existence of an abelian group with all non-zero elements transcendental.

Abstract

It is well known that value at a non-zero algebraic number of each of the functions $e^{x}, \ln x, \sin x, \cos x, \tan x, \csc x, \sec x, \cot x, \sinh x,$ $ \cosh x,$ $ \tanh x,$ and $\coth x$ is transcendental number (see Theorem 9.11 of \cite{N}). In the work, we show that for any one of the above mentioned functions, $f(x)$, and for a polynomial $g(x)$ with rational coefficients the zero, if any, of the equation $f(x)=g(x)$ is a transcendental number. We also show that if $f(x)$ and $g(x)$ are polynomials with rational coefficients, then a zero of the equation $e^{f(x)}=g(x)$ is a transcendental number. Finally we show that the existence of an abelian group whose non-zero elements are transcendental numbers.