Algebraic values of transcendental power series with geometric coefficient moduli

Abstract

Let $\lambda >1$ be a real algebraic number. We construct continuum many power series $f(z)={\sum }_{k\ge 0}{a}_k{z}^k$ of radius of convergence exactly one such that every nonzero coefficient $a_k$ is algebraic and has modulus ${\lambda }^m$ for some $m\ge 0$. Moreover, for every integer $s\ge 0$, 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.