Solvability of meromorphic equations in elementary functions

TL;DR

This paper investigates the solvability of complex meromorphic equations in elementary functions, proving that many such functions with certain properties are unsolvable in elementary functions using topological Galois theory.

Contribution

It generalizes previous results by proving the unsolvability of all elementary meromorphic functions with infinitely many roots of their derivatives and infinitely many distinct values at those roots.

Findings

Many elementary meromorphic functions are unsolvable in elementary functions.

The proof extends topological Galois theory to a broader class of functions.

Almost all entire surjective functions of exponential growth are covered by previous results.

Abstract

An equation $f(x)=a$, where $f$ is a complex meromorphic function and $a\in\mathbb{C}$ is a parameter, is solvable in elementary functions if the inverse map $x=f^{-1}(a)$ can be expressed as a finite composition of arithmetic operations (addition, subtraction, multiplication, and division), the exponential function, the complex logarithm, and constants. Specific functions such as $\tan x - x$, $\exp x + x$, $x^x$ have been proven to be unsolvable by Kanel-Belov, Malistov, Zaytsev, while almost all entire surjective functions of at most exponential growth have been covered by Zelenko. All these rely on one-dimensional topological Galois theory, developed by Khovanskii. We generalize to provide a proof for the unsolvability of all elementary meromorphic functions $f$ such that the derivative of $f$ has infinitely many roots $x_i$ and the set of distinct values $f(x_i)$ is infinite.