Algebraic values of transcendental power series with geometric coefficient moduli

Abstract

Let $λ>1$ be a real algebraic number. We construct continuum many power series $f(z)=\sum_{k\geq0}a_kz^k$ of radius of convergence exactly one such that every nonzero coefficient $a_k$ is algebraic and has modulus $λ^m$ for some $m\geq0$. Moreover, for every integer $s\geq0$, the derivative $f^{(s)}$ takes algebraic values at all algebraic points of the open unit disk and is transcendental over $\mathbb{C}(z)$. The proof combines algebraic polygonal cancellation with a sparse polynomial-block argument. This shows that a multiplicative rank-one restriction on coefficient moduli is compatible with algebraicity of the full analytic jet at every algebraic point once algebraic phases are allowed.

Disclosure

“Declaration of competing interest The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the author used ChatGPT (OpenAI) solely for English grammar correction and language editing. The author reviewed t”

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2026-08-23T19:09:39.375940Z

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