On deciding transcendence of power series

2025/04/23 by Alin Bostan, Bruno Salvy, Bostan, Alin +3 · 3 voices · 4 citations
Computer Science · Mathematics · #11J81 #16S32 #34M15 #Classical Analysis and ODEs (math.CA) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #cs.SC #math.CA #math.NT

paper · pdf · doi:10.48550/arxiv.2504.16697

openalex publication_date 2025/04/23 · arxiv published 2025/04/23 · arxiv updated 2025/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The diagonal and Hadamard grade of hypergeometric functions
• On the diagonals of rational functions: the minimal number of variables (unabridged version)
• Algebraicity of hypergeometric functions with arbitrary parameters
• Algebraicity of hypergeometric functions with arbitrary parameters
• Algebraic solutions of linear differential equations: an arithmetic approach
• Minimization of differential equations and algebraic values of\n E-functions
• Symbolic-Numeric Factorization of Differential Operators
• Solutions of algebraic linear ordinary differential equations
• Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz
• Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems
• Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior
• Degree bound for toric envelope of a linear algebraic group
• A New Bound on Hrushovski's Algorithm for Computing the Galois Group of a Linear Differential Equation
• Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane
• Lattice Green Functions: the d-dimensional face-centred cubic lattice, d=8, 9, 10, 11, 12
• Lattice Green Functions: the seven-dimensional face-centred cubic lattice
• Hrushovski's Algorithm for Computing the Galois Group of a Linear Differential Equation
• Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices
• Power Series Solutions of Singular (q)-Differential Equations
• Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First
• A Fast Approach to Creative Telescoping
• Lattice Green functions in all dimensions
• The complete Generating Function for Gessel Walks is Algebraic
• Degenerate Gauss hypergeometric functions
• Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité
• The solution of the problem of integration in finite terms
• Über einige von Herrn Dr. Eisenstein aufgestellte Lehrsätze, irreductible Congruenzen betreffend. (S. 182 Bd. 39 dieses Journals).
• Von denjenigen Moduln, welche Potenzen von Primzahlen sind.(Fortsetzung)

Cited by

• On the Disk of Convergence of Algebraic Power Series
• Symmetric Division of Linear Ordinary Differential Operators
• An Effective Version of the p-Curvature Conjecture for Order One Differential Equations

Discussions

• #arXiv On deciding transcendence of power series https://arxiv.org/abs/2504.16697 “Given D-finite power series, is it algebraic or transcendental? This problem is decidable: two theoretical algorithms [bsky, 2 points, 0 comments]
• #arXiv On deciding transcendence of power series https://arxiv.org/abs/2504.16697 “Given D-finite power series, is it algebraic or transcendental? This problem is decidable: two theoretical algorithms [bsky, 0 points, 0 comments]
• # arXiv On deciding transcendence of power series https:// arxiv.org/abs/2504.16697 “Given D-finite power series, is it algebraic or transcendental? This problem is decidable: two theoretical algorith [mastodon, 0 points, 0 comments]

Related

• Finiteness results concerning algebraic power series
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• An arithmetic characterization of some algebraic functions and a new proof of an algebraicity prediction by Golyshev
• Single-exponential bounds for diagonals of D-finite power series
• On Existence and Uniqueness of Formal Power Series Solutions of Algebraic Ordinary Differential Equations