The depth-one decomposition hypothesis for solvable meromorphic equations

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Let X⊆CX\subseteq\mathbb{C} be a connected open set and let f:X→Cf:X\to\mathbb{C} be a meromorphic elementary function. A function is solvable when its equation with a constant right-hand side is solvable in elementary functions, and its depth is the notion used in the paper. Depth-one decomposition hypothesis. If the equation f(x)=af(x)=a is solvable in elementary functions, then there exists a finite sequence g1,g2,…,gℓg_1,g_2,\ldots,g_\ell of solvable functions of depth at most 11 such that

f(x)=(g1∘g2∘…∘gℓ)(x).f(x)=(g_1\circ g_2\circ\ldots\circ g_\ell)(x).

This hypothesis proposes that every elementary-solvable meromorphic equation admits a decomposition into solvable factors whose depth is at most one. The source presents it as a hypothesis based on modest computational evidence; its status is therefore open.

References

Primary source

Miroslav Marinov and Nikola Veselinov, “Solvability of meromorphic equations in elementary functions”, arXiv:2602.09253 (2026).

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