Solvability of meromorphic equations in elementary functions
Miroslav Marinov, Nikola Veselinov
TL;DR
The paper investigates when equations of the form $f(x)=a$, with $f$ a complex meromorphic elementary function, admit an explicit solution in terms of elementary operations. It leverages one-dimensional topological Galois theory to show that solvability is governed by the monodromy group of the inverse branches of $f$, with Wielandt's theorem forcing unsolvability whenever the monodromy includes a nontrivial infinite action. The main contribution is a general unsolvability result: if $f$ has infinitely many critical values (i.e., infinite branch locus), then $f(x)=a$ is unsolvable in elementary functions, and this persists through finite decompositions into factors with primitive monodromy. The work extends prior results for specific functions to a broad class of elementary meromorphic functions, clarifying structural obstructions to explicit solvability and proposing a conjectured criterion that solvability would require a finite chain of depth-1 solvable factors, i.e., a highly restrictive form.
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.