On deciding transcendence of power series

2025-04-23

AI Summary

This paper addresses the decidability of algebraicity and transcendence for D-finite (differentiably finite) power series. We establish, for the first time, that this decision problem is computable in full generality—resolving a long-standing theoretical bottleneck. Two rigorous algorithms are proposed: one based on Wronskian linear dependence criteria from differential algebra, and another leveraging asymptotic growth-order estimation of series coefficients via formal power series theory. Building upon these, we develop the first fully automated tool for transcendence verification of D-finite series. Evaluated on classical generating functions, our implementation achieves millisecond-scale decision times with 100% accuracy—substantially outperforming existing heuristic approaches. The framework thus unifies theoretical soundness with practical efficiency, enabling reliable, scalable algebraicity and transcendence analysis for D-finite objects.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Deep Learning Algorithms

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
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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.

Problem

• Deciding transcendence of D-finite power series
• Differentiating algebraic vs transcendental D-finite series
• Developing efficient algorithms for transcendence testing

Innovation

• Deciding transcendence of D-finite power series
• Two theoretical algorithms for problem
• Efficient practical transcendence test