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Algebraic and Arithmetic Attributes of Hypergeometric Functions in SageMath

This paper presents SageMath implementations of algorithms for analyzing the algebraic and arithmetic properties of hypergeometric functions over rational numbers, finite fields, and p-adic fields, including capabilities to decide algebraicity, compute valuations, and determine minimal polynomials in positive characteristic.

Original authors: Xavier Caruso, Florian Fürnsinn

Published 2026-02-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xavier Caruso, Florian Fürnsinn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical recipe book. In this book, the recipes aren't for cakes or soups, but for Hypergeometric Functions. These are complex mathematical formulas that look like infinite lists of numbers added together (power series). They are famous in the math world because they pop up everywhere, from physics to combinatorics.

The authors of this paper, Xavier Caruso and Florian Fürnsinn, have built a new set of digital tools inside a software called SageMath to help mathematicians understand these recipes better. Think of SageMath as a high-tech kitchen, and their new package as a specialized set of measuring cups, scales, and ovens designed specifically for these tricky recipes.

Here is a breakdown of what their new tools can do, explained through simple analogies:

1. The Three Kitchens (Where the Recipes Live)

The paper explains that these mathematical recipes can be cooked in three different "kitchens" (mathematical environments), and their new tools work in all of them:

The Rational Kitchen ( $\mathbb{Q}$ ): The standard kitchen using normal fractions (like 1/3 or 2/5).
The Finite Field Kitchen ( $\mathbb{F}_p$ ): A kitchen with a limited number of ingredients, like a clock that only has 12 hours (or 13, or 17). You wrap around when you go past the limit.
The $p$ -adic Kitchen ( $\mathbb{Q}_p$ ): A strange, abstract kitchen where "closeness" is measured differently. Here, numbers are close if their difference is divisible by a large power of a prime number (like 5 or 3).

2. Checking the Recipe's Nature (Algebraic & Arithmetic Properties)

Before cooking, you want to know what kind of dish you're making. The new tools can answer specific questions:

Is it "Globally Bounded"?
- Analogy: Imagine a recipe that, no matter how many times you scale it up, never produces a number so huge it breaks the calculator. The tool checks if the recipe stays "well-behaved" everywhere.
Is it "Algebraic"?
- Analogy: Some recipes are simple enough that they can be described by a single, finite equation (like $y = x^2 + 1$ ). Others are so complex they can't be. The tool decides: "Yes, this one is simple," or "No, this one is infinitely complex."
Good Reduction (The "Mod $p$ " Test):
- Analogy: Imagine taking your recipe and trying to cook it in the "Finite Field Kitchen" (the clock kitchen). Sometimes, the ingredients don't work (you get a division by zero error). The tool tells you exactly which "clock sizes" (prime numbers) allow the recipe to work without breaking.

3. The "Section" and "Dwork" Magic (Breaking it Down)

When working in the Finite Field Kitchen, the tools use a special trick called Section Operators.

Analogy: Imagine you have a long, winding river (the infinite series). The tool cuts the river into small, manageable segments (sections). It turns out that for these specific recipes, every segment is just a simple copy of the original river, maybe stretched or shrunk a bit.
Dwork Relations: The tool uses these segments to write the original recipe as a combination of other, simpler recipes raised to a power. It's like saying, "This complex stew is actually just a mix of three simpler soups, cooked at high heat."

4. Finding the "Kill Switch" (Annihilating Polynomials)

Every complex recipe has a "kill switch"—a specific mathematical operation that, if applied, turns the whole thing into zero.

Analogy: The tool finds the exact "off switch" for the recipe. It writes down a polynomial (a mathematical formula) that, when applied to the function, makes it vanish. This is crucial for proving properties about the function.

5. Comparing Recipes (Congruences)

Sometimes, two recipes with different ingredients end up tasting exactly the same in the Finite Field Kitchen.

Analogy: The tool can compare two different recipes and say, "Hey, even though these look different, if you cook them in a 13-hour clock kitchen, they produce the exact same result." It does this by checking tiny slices of the recipes recursively, like tasting a spoonful from the start, middle, and end to see if they match.

6. The $p$ -adic Kitchen (Valuations and Convergence)

In the strange $p$ -adic kitchen, the rules of distance change.

Radius of Convergence: This is the "safe zone." If you try to cook the recipe with an ingredient that is too far away (too large), the dish explodes (diverges). The tool calculates exactly how far you can go before it explodes.
Valuations: This measures the "cleanliness" of the ingredients. In this kitchen, a number is "cleaner" if it is divisible by a high power of the prime number. The tool tells you the "cleanliness level" of the final dish.
Newton Polygons: This is a visual map (a graph) that shows the "terrain" of the recipe's ingredients. It helps mathematicians see the shape of the function's behavior. The tool draws this map, even if the terrain goes on forever, by cutting it off at a safe distance.

Summary

In short, Caruso and Fürnsinn have built a Swiss Army knife for Hypergeometric Functions. Before this, mathematicians had to do these complex checks by hand or with very limited tools. Now, they can use SageMath to instantly check if a function is algebraic, see how it behaves in different mathematical "worlds," find its kill switches, and visualize its structure.

The paper doesn't claim these tools will cure diseases or build bridges directly; rather, it provides the fundamental testing equipment that allows mathematicians to formulate and check deep theories about how these numbers behave. It's about giving scientists better microscopes to look at the structure of mathematics itself.

Technical Summary: Algebraic and Arithmetic Attributes of Hypergeometric Functions in SageMath

Problem Statement
Hypergeometric functions, defined as power series with rational parameters, are central objects in the study of D-finite series. While their algebraic and arithmetic properties—such as algebraicity, global boundedness, and behavior under reduction modulo primes—are theoretically classified, there is a practical need for computational tools to formulate and verify conjectures regarding these properties. Specifically, researchers require software capable of handling hypergeometric series over rational numbers ( $\mathbb{Q}$ ), finite fields ( $\mathbb{F}_p$ ), and $p$ -adic fields ( $\mathbb{Q}_p$ ) to test hypotheses about Galois groups of reductions and other arithmetic phenomena. The authors identified a gap in existing computer algebra systems regarding the implementation of these specific algorithms within a unified framework.

Methodology
The authors developed a new SageMath package that implements hypergeometric functions as elements of symbolic rings, allowing for manipulation over arbitrary rings (rational fields, polynomial rings, power series rings, and $p$ -adic fields). The implementation relies on several distinct algorithmic approaches tailored to the specific arithmetic context:

Over $\mathbb{Q}$ : The package implements criteria for global boundedness and algebraicity. These rely on the relative positions of the decimal parts of the parameters (Christol's criterion) and extensions for cases involving integer differences between top and bottom parameters. The system also computes sets of primes with good reduction by analyzing the $p$ -adic valuation of coefficients for small primes and extrapolating based on congruence classes modulo the least common denominator of the parameters.
Over $\mathbb{F}_p$ : For series with good reduction, the package utilizes section operators to decompose hypergeometric functions. It constructs Dwork relations, expressing a function as a polynomial linear combination of $p$ -th powers of other hypergeometric functions. These relations are used to compute annihilating polynomials (linearized polynomials in the Frobenius operator). Furthermore, the package includes an algorithm to check if two distinct hypergeometric series are congruent modulo $p$ by recursively comparing their sections and constant terms. For functions where $m=n-1$ , the package computes the $p$ -curvature of the associated differential operator.
Over $\mathbb{Q}_p$ : The implementation focuses on convergence and valuation. It calculates the logarithmic $p$ -adic radius of convergence using an explicit formula based on the partition of parameters into $p$ -adic integers and non-integers. It computes $\nu$ -drifted $p$ -adic valuations using a recursion over the tropical semi-ring and the Floyd-Warshall algorithm to determine the weak transitive closure of a tropical matrix. Finally, it computes Newton polygons and evaluates the functions at specific $p$ -adic points within their radius of convergence.

Key Contributions

Unified Implementation: The creation of an open-source SageMath package that treats hypergeometric functions over $\mathbb{Q}$ , $\mathbb{F}_p$ , and $\mathbb{Q}_p$ within a single interface.
Algorithmic Realization: The translation of theoretical classifications (e.g., Christol's criteria, Beukers-Heckman classification) and arithmetic properties (Dwork relations, $p$ -curvature) into executable code.
Specific Algorithms:
- is_globally_bounded() and is_algebraic() for rational parameters.
- good_reduction_primes() to determine primes where coefficients have denominators coprime to $p$ .
- section(), dwork_relation(), and annihilating_ore_polynomial() for arithmetic properties in positive characteristic.
- is_equal_as_series() to detect congruences between different hypergeometric series modulo $p$ .
- p_curvature() and p_curvature_coranks() for analyzing differential operators in finite fields.
- log_radius_of_convergence(), valuation(), and newton_polygon() for $p$ -adic analysis.
Handling Edge Cases: The system accommodates general parameters, including nonnegative integers in bottom parameters if compensated by top parameters, and handles cases where algorithms might theoretically involve infinite systems (with a warning that termination is not strictly proven for the simplified implementation of annihilating polynomials).

Results
The paper demonstrates the functionality of the package through various examples:

Rational Case: Successfully classifies functions $f$ , $g$ , and $h$ as globally bounded or algebraic and identifies specific congruence classes of primes for good reduction.
Finite Field Case: Computes Dwork relations for a function modulo 19, deriving a dictionary of polynomial coefficients. It demonstrates the computation of annihilating polynomials and verifies the congruence of two different hypergeometric series modulo 13. It also calculates the $p$ -curvature matrix and its coranks for various primes.
$p$ -adic Case: Computes the logarithmic radius of convergence (e.g., $-7/2$ for a specific function at $p=5$ ), calculates drifted valuations (returning tuples of valuation and minimal index), and constructs Newton polygons, successfully handling cases with infinite slopes by truncation.

Significance and Claims
The authors position this work as a practical tool for researchers working on conjectures concerning the algebraic and arithmetic properties of D-finite series, a need that arose during their joint work on Galois groups of reductions modulo primes. The paper claims that the package provides the necessary functionalities to manipulate hypergeometric functions in these diverse settings, facilitating the testing of theoretical predictions.

The authors are modest regarding the theoretical completeness of the implementation; they explicitly warn that the algorithm for computing annihilating polynomials uses a simplified version where the finiteness of the involved hypergeometric series is not proven, meaning computations might not terminate in all cases. However, they express confidence that the current implementation will terminate in practice. The work is presented as a step toward integration into a future release of SageMath, complemented by an interactive worksheet for exploration.