Algebraic values of transcendental power series with geometric coefficient moduli

2026/07/26 by Diego Marques
#math.NT

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Abstract

Let λ>1 be a real algebraic number. We construct continuum many power series f(z)=∑_k≥0a_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≥0. Moreover, for every integer s≥0, the derivative f^(s) takes algebraic values at all algebraic points of the open unit disk and is transcendental over 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.

Citations

• Values of generalized Liouville power series at algebraic numbers
• A positive answer for a question proposed by K. Mahler
• Algebraic Values of Transcendental Functions at Algebraic Points

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