Algebraic and Arithmetic Attributes of Hypergeometric Functions in SageMath

We report on implementations for algorithms treating algebraic and arithmetic properties of hypergeometric functions in the computer algebra system SageMath. We treat hypergeometric series over the rational numbers, over finite fields, and over the p-adics. Among other things, we provide implementations deciding algebraicity, computing valuations, and computing minimal polynomials in positive characteristic.

💡 Research Summary

The paper presents a comprehensive suite of algorithms implemented in SageMath for investigating the algebraic and arithmetic properties of generalized hypergeometric functions nF_m(α,β;x). The authors treat three ambient coefficient rings: the rational numbers ℚ, finite fields 𝔽_p, and p‑adic fields ℚ_p. For each setting they provide concrete SageMath methods that decide global boundedness, algebraicity, good reduction at primes, p‑adic valuations, radii of convergence, minimal annihilating polynomials, and related invariants.

1. Implementation Overview
Hypergeometric series are introduced as symbolic objects in SageMath via the constructor hypergeometric(