Solvability of Meromorphic Equations in Elementary Functions

Overview

An equation f(x)=a, where f is a complex meromorphic function and a 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, the exponential function, and the complex logarithm. Using methods from topological Galois theory, Kanel-Belov, Malistov and Zaytsev proved unsolvability for the equations tan x-x=a and x^x=a (and its equivalent exp x+x=a), while Zelenko covered almost all entire surjective functions of at most exponential growth. We generalize these results to prove that if the derivative of f has infinitely many roots x_i and the set of distinct values f(x_i) is infinite, then f(x)=a is unsolvable in elementary functions. The proof combines Wielandt's theorem on primitive actions of infinite permutation groups with an iterative decomposition of the monodromy group into primitive factors via one-dimensional topological Galois theory.

Awards (5)

• First Award of $6,000 $6,000
• Regeneron Young Scientist Awards
• American Mathematical Society: One-Year Membership to American Mathematical Society to each winner (7 winning projects, up to 3 team members per project)
• American Mathematical Society: Second Award of $1,000 $1,000
• Mu Alpha Theta, National High School and Two-Year College Mathematics Honor Society: First Award of $ 1,500 $1,500

Competition history

• ISEF 2026 Mathematics · Entry MATH006

Resources

• ISEF abstract page ↗

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