初等函数中解析方程的可解性
Solvability of meromorphic equations in elementary functions
AI总结：
本文基于一维拓扑Galois理论，证明了所有导数有无限多个根且值域无限的初等解析函数在初等函数中不可解。
摘要：
一个方程$f(x)=a$，其中$f$是一个复解析函数，$a\in\mathbb{C}$是一个参数，如果逆映射$x=f^{-1}(a)$可以表示为有限次算术运算（加法、减法、乘法和除法）、指数函数、复对数和常数的组合，则称该方程在初等函数中可解。特定函数如$\tan x - x$、$\exp x + x$、$x^x$已被Kanel-Belov、Malistov、Zaytsev证明不可解，而几乎所有增长不超过指数的整个满射函数已被Zelenko覆盖。所有这些都依赖于Khovanskii发展的一维拓扑Galois理论。我们推广这一理论，证明所有初等解析函数$f$，其导数有无限多个根$x_i$且$f(x_i)$的值域无限的不可解性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
