On deciding transcendence of power series
Alin Bostan, Bruno Salvy, Michael F. Singer
TL;DR
This work proves the decidability of whether a D-finite power series $f\in\mathbb{Q}[[z]]$ is algebraic or transcendental, introducing two theoretical algorithms and a practical transcendence test that leverage the minimal annihilator $L^{\min}_f$. It develops two complementary strategies: (i) a Stanley-style algorithm for (S) based on $L^{\min}_f$ and Singer’s framework, and (ii) an algorithm for (P) computing the algebraic-right-factor $L^{\mathrm{alg}}$ using differential Galois theory and LCLM constructions; both rely on the interplay between $L^{\min}_f$, algebraic solutions, and diagonals. The paper further discusses conditional enhancements in the globally bounded case (Stanley2b) under Christol–Andre-type conjectures and surveys alternative approaches (p-curvatures, guess-and-prove, factoring). A rich set of examples—Apéry-type series, diagonals, and lattice Green functions—demonstrates the practical effectiveness of the methods and highlights their relevance to combinatorics and number theory. Collectively, the results establish a principled, implementable path toward deciding algebraicity versus transcendence for a broad class of D-finite functions and illuminate connections to hypergeometric theory and differential Galois theory.
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.