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Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions

This paper presents algorithms to compute the p-adic valuation of hypergeometric functions on specific disks, determine the set of primes modulo which they can be reduced, and find annihilating polynomials for their reductions.

Original authors: Xavier Caruso, Florian Fürnsinn

Published 2026-02-06

📖 6 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

The Big Picture: The Infinite Recipe

Imagine you have a recipe for a cake that never ends. You keep adding ingredients forever. In math, this is called a hypergeometric function. It's a specific type of infinite recipe (or power series) defined by a set of numbers (parameters) that tell you exactly how much of each ingredient to add at every step.

Usually, these recipes use complex numbers or fractions. But this paper asks a very specific question: What happens if we try to bake this cake using only a limited set of ingredients, specifically looking at the "flavor" of the numbers when divided by a prime number (like 2, 3, 5, 7)?

The authors, Xavier Caruso and Florian Fürnsinn, have built a set of algorithms (step-by-step computer instructions) to answer three main questions about these infinite recipes:

Can we bake it at all? (Does the recipe work when reduced modulo a prime?)
How "deep" does the flavor go? (What is the $p$ -adic valuation?)
Is there a hidden rule? (Can we find a simple equation that describes the recipe's behavior?)

1. The "Valuation" Detective: Measuring the Flavor Depth

To understand if a recipe works with a specific prime number (let's call it $p$ ), the authors look at the "valuation" of the ingredients.

The Analogy: Imagine you are counting how many times you can divide a number by $p$ before you get a fraction.

If you have the number 8 and $p=2$ , you can divide by 2 three times ( $8 \to 4 \to 2 \to 1$ ). The valuation is 3.
If you have the number 7 and $p=2$ , you can't divide it at all. The valuation is 0.
If you have a fraction like $1/2$ , the valuation is negative (it's "too small" to be a whole number).

The Problem: The recipe has infinite steps. Sometimes, as you go further down the list of ingredients, the "divisibility by $p$ " gets worse and worse (the valuation goes to negative infinity). If this happens, the recipe breaks when you try to reduce it modulo $p$ .

The Solution: The authors created a Zigzag Function.
Imagine a hiker walking up and down a mountain range. The height of the mountain at any point represents the valuation of the recipe's ingredient at that step.

The authors realized this mountain range isn't random; it follows a strict, repeating pattern (a "zigzag").
They developed a way to calculate the lowest point of this mountain range.
The Result: If the lowest point is above sea level (non-negative), the recipe works perfectly for that prime. If the lowest point dives below sea level, the recipe breaks.

They also figured out how to calculate the Newton Polygon. Think of this as a map of the mountain range's shape. It tells you not just the lowest point, but the entire landscape of how the ingredients behave, which helps predict if the recipe will converge (stop making sense) or explode.

2. The "Prime Filter": Who Can We Use?

Not every prime number is friendly to every recipe. Some primes cause the recipe to collapse immediately.

The Analogy: Imagine you have a sieve (a filter) with holes of different sizes. You want to know which grains (primes) can pass through the sieve without getting stuck.

The authors found that for most large primes, the behavior of the recipe depends only on the remainder the prime leaves when divided by a specific number (the common denominator of the recipe's parameters).
It's like saying: "If the prime number is 1 more than a multiple of 3, the recipe works. If it's 2 more, it fails."

The Algorithm: They built a tool that scans through all possible prime numbers and sorts them into two buckets:

Good Primes: The recipe works fine.
Bad Primes: The recipe breaks.
They proved that the "Good Primes" form a predictable pattern (arithmetic progressions), making it easy for a computer to list them all.

3. The "Shrinking Machine": Finding the Hidden Rule

This is the most magical part of the paper.

The Analogy: Imagine you have a giant, complex machine (the infinite recipe) that spits out numbers. You want to find a simple rule that explains how the machine works.

Usually, infinite recipes are too messy to have a simple rule.
However, the authors discovered that if you look at the recipe modulo a prime (using only the "flavor" of the numbers), the machine suddenly becomes much simpler. It turns out to be algebraic.

What does "Algebraic" mean here?
It means the infinite recipe is actually just a solution to a simple polynomial equation (like $y^2 + 3y + 1 = 0$ ). Even though the recipe goes on forever, it is "tied down" by a simple mathematical leash.

How they found it:
They used a tool called the Dwork Map and Section Operators.

Think of the Section Operator as a machine that takes your infinite recipe and cuts out every $p$ -th ingredient, creating a new, shorter recipe.
The authors proved that if you keep cutting and shrinking the recipe, you eventually run out of new variations. You only get a finite number of different "versions" of the recipe.
Because there are only a finite number of versions, they can be arranged into a system of equations. Solving this system gives you the annihilating polynomial—the simple leash that holds the infinite recipe.

Summary of Their Contributions

The Valuation Calculator: A fast way to determine if a hypergeometric recipe works for a specific prime number by analyzing its "mountain range" of values.
The Prime Scanner: A method to list exactly which prime numbers allow the recipe to work, showing that these primes follow a predictable pattern.
The Algebraic Finder: A guaranteed method to find the simple polynomial equation that describes the recipe when reduced modulo a prime. This proves that even though the recipe is infinite, its "modulo $p$ " version is mathematically simple and structured.

Why does this matter?
The paper doesn't just say "it's possible"; it gives the computer code (implemented in SageMath) to do it. They handle both "large" primes (where things are easy) and "small" primes (where things are tricky and require careful handling), ensuring the tools work for every case.

In short, they built a toolkit to take a messy, infinite mathematical object, filter it through the lens of prime numbers, and reveal the simple, structured rules hiding underneath.

Technical Summary: Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions

Problem Statement
The paper addresses the computational challenges associated with the arithmetic properties of hypergeometric series $H(\boldsymbol{\alpha}, \boldsymbol{\beta}; x)$ with rational parameters. While the algebraic nature of such series over $\mathbb{Q}$ is well-understood (occurring only for specific parameter sets), their behavior modulo a prime $p$ is more complex. Specifically, the authors aim to develop algorithmic tools to:

Determine the set of prime numbers for which a given hypergeometric function admits a "good reduction" (i.e., all coefficients lie in the $p$ -adic integers $\mathbb{Z}_{(p)}$ ).
Compute the $p$ -adic valuation of the series on disks within the radius of convergence.
Construct an annihilating polynomial for the reduction of the series modulo $p$ , thereby proving its algebraicity over $\mathbb{F}_p(x)$ without restrictive assumptions on parameters or the prime $p$ .

Methodology
The authors employ a combination of $p$ -adic analysis, tropical algebra, and the theory of Dwork maps. The core methodology is structured as follows:

$p$ -adic Valuation via Zigzag Functions: The valuation of the coefficients $h_k$ is analyzed using "zigzag functions" $w_r(k)$ , which count the occurrences of parameters congruent to $0 \pmod{p^r}$ . The total valuation is expressed as a sum of these functions (Lemma 2.1).
Drifted Valuations and Tropical Recurrence: To handle the valuation uniformly, the authors introduce "drifted valuations" $\text{val}_{p, \nu}(h(x)) = \min_k (\text{val}_p(h_k) + \nu k)$ . They model the computation of these valuations using the tropical semiring (where addition is $\min$ and multiplication is $+$ ). This allows the valuation problem to be reformulated as a recurrence relation involving matrices over the tropical semiring.
Newton Polygons: The authors extend the tropical approach to compute the Newton polygon of the series. By utilizing a semiring of closed convex sets (isomorphic to the semiring of concave functions via the Legendre transform), they derive a uniform algorithm to compute the polygon, which encodes the valuations for all drifts $\nu$ .
Section Operators and Dwork Maps: To study the reduction modulo $p$ , the paper utilizes section operators $\Lambda_r$ (extracting coefficients $a_{kp+r}$ ) and the Dwork map $\mathfrak{D}_p$ . Proposition 3.5 establishes that the $r$ -th section of a hypergeometric series is multiplicatively congruent to a scalar multiple of another hypergeometric series with transformed parameters.
Iterative Reduction: By iterating the Dwork relations, the authors show that the reduction of the series modulo $p$ can be expressed as a linear combination of $p$ -th powers of a finite set of auxiliary hypergeometric series.

Key Contributions and Results

Algorithm for Good Reduction Primes:
The authors provide an algorithm to determine the set of primes $P_h$ for which $H(\boldsymbol{\alpha}, \boldsymbol{\beta}; x)$ has good reduction. They prove that for sufficiently large primes (specifically $p > \max(B(\boldsymbol{\alpha}, \boldsymbol{\beta}), 2m)$ ), the property of having good reduction depends only on the congruence class of $p$ modulo a common denominator $d$ of the parameters. This reduces the problem to checking a finite set of primes and congruence classes.
Computation of $p$ -adic Valuations and Newton Polygons:
The paper presents efficient algorithms to compute the $p$ -adic valuation of the series on any disk of radius smaller than the radius of convergence.
- For the case where the drift $\nu$ equals the critical value $\nu_0$ , the algorithm relies on the periodicity of the transition matrices in the tropical semiring.
- For $\nu > \nu_0$ , the algorithm utilizes a halting criterion based on the growth of valuations to terminate the recurrence.
- These methods are extended to compute the full Newton polygon of the series, handling cases where the polygon has infinitely many slopes by restricting the domain of definition.
Annihilating Polynomials Modulo $p$ :
Building on the work of Christol, Vargas-Montoya, and the authors, the paper describes an algorithm to compute an annihilating polynomial for $H(\boldsymbol{\alpha}, \boldsymbol{\beta}; x) \pmod p$ .
- The method constructs a finite set $Y$ of parameter tuples generated by the Dwork map.
- It establishes a linear relation between the powers of the series and the $p$ -th powers of series associated with parameters in $Y$ .
- By solving a linear system over $\mathbb{F}_p[x]$ , the authors derive a non-trivial polynomial $P(x, y)$ such that $P(x, H(\boldsymbol{\alpha}, \boldsymbol{\beta}; x)) \equiv 0 \pmod p$ .

Significance and Claims
The authors claim that their approach offers a new, unconditional proof of the algebraicity of hypergeometric series modulo $p$ . Unlike previous works that often excluded specific cases for simplicity (e.g., requiring large primes or specific parameter constraints), this algorithm works for any prime $p$ and any rational parameters, provided the series is well-defined.

The paper emphasizes the importance of handling "small primes" explicitly. While large primes often lead to computationally intractable outputs (such as annihilating polynomials with huge coefficients), the proposed algorithms are designed to treat small primes effectively, which is crucial for practical applications where extracting relevant information from massive outputs is difficult.

Finally, the authors note that all algorithms have been implemented in SageMath and are available for testing, providing a practical tool for researchers investigating the arithmetic properties of D-finite series and Christol's conjecture.

The Big Picture: The Infinite Recipe

1. The "Valuation" Detective: Measuring the Flavor Depth

2. The "Prime Filter": Who Can We Use?

3. The "Shrinking Machine": Finding the Hidden Rule

Summary of Their Contributions

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