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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.09253 |
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Table of Contents:
- 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.