Algebraic and Arithmetic Attributes of Hypergeometric Functions in SageMath

TL;DR

The paper presents a SageMath package that implements algorithms for the algebraic and arithmetic properties of hypergeometric functions across $\mathbb{Q}$, $\mathbb{F}_p$, and $\mathbb{Q}_p$. It introduces concrete methods for testing global boundedness and algebraicity (via Christol-type criteria), identifying primes with good reductions, and analyzing reductions modulo primes through Dwork relations, annihilating polynomials, and congruences. It also develops $p$-adic tools, including radius of convergence, $p$-adic valuations, $p$-adic evaluations, and Newton polygons, to study convergence and arithmetic properties in the $p$-adic setting. Together these contributions provide practical, open-source computational tools and an interactive notebook for testing conjectures about D-finite series and their arithmetic behavior.

Abstract

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.