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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.09253 |
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| _version_ | 1866912893619929088 |
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| author | Marinov, Miroslav Veselinov, Nikola |
| author_facet | Marinov, Miroslav Veselinov, Nikola |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solvability of meromorphic equations in elementary functions Marinov, Miroslav Veselinov, Nikola Group Theory Complex Variables 20B27, 20F16 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. |
| title | Solvability of meromorphic equations in elementary functions |
| topic | Group Theory Complex Variables 20B27, 20F16 |
| url | https://arxiv.org/abs/2602.09253 |