On deciding transcendence of power series

Alin Bostan; Bruno Salvy; Michael F. Singer

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arXiv · 2504.16697

On deciding transcendence of power series

Alin Bostan·Bruno Salvy·Michael F. Singer

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Abstract

It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem, whether a given D-finite power series is algebraic or transcendental, is notoriously difficult. We prove that this problem is decidable: we give two theoretical algorithms and a transcendence test that is efficient in practice.

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Alin Bostan, Bruno Salvy, Michael F. Singer. 2025-04-23. On deciding transcendence of power series. https://arxiv.org/abs/2504.16697

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