Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions

Xavier Caruso, Florian Fürnsinn

TL;DR

The paper develops comprehensive algorithms for understanding arithmetic properties of hypergeometric functions, notably $p$-adic valuations, Newton polygons, and reductions modulo primes. It combines zigzag and drifted-valuation techniques with tropical and tropicalized–Dwork frameworks to compute valuations, Newton polygons, and $p$-adic evaluations, and to establish algebraicity modulo $p$ via section operators and Dwork relations. A key contribution is an automatic construction of annihilating polynomials for reductions $h(x)\bmod p$, together with a clear analysis of good reduction primes and their dependence on congruence classes, all implemented in SageMath. The results advance both theoretical understanding and practical computation of algebraicity and reductions of hypergeometric series, with implications for Christol-type conjectures and Diophantine properties of these functions.

Abstract

We discuss algorithms for arithmetic properties of hypergeometric functions. Most notably, we are able to compute the p-adic valuation of a hypergeometric function on any disk of radius smaller than the p-adic radius of convergence. This we use, building on work of Christol, to determine the set of prime numbers modulo which it can be reduced. Moreover, we describe an algorithm to find an annihilating polynomial of the reduction of a hypergeometric function modulo p.

Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions

TL;DR

The paper develops comprehensive algorithms for understanding arithmetic properties of hypergeometric functions, notably -adic valuations, Newton polygons, and reductions modulo primes. It combines zigzag and drifted-valuation techniques with tropical and tropicalized–Dwork frameworks to compute valuations, Newton polygons, and -adic evaluations, and to establish algebraicity modulo via section operators and Dwork relations. A key contribution is an automatic construction of annihilating polynomials for reductions , together with a clear analysis of good reduction primes and their dependence on congruence classes, all implemented in SageMath. The results advance both theoretical understanding and practical computation of algebraicity and reductions of hypergeometric series, with implications for Christol-type conjectures and Diophantine properties of these functions.

Abstract

We discuss algorithms for arithmetic properties of hypergeometric functions. Most notably, we are able to compute the p-adic valuation of a hypergeometric function on any disk of radius smaller than the p-adic radius of convergence. This we use, building on work of Christol, to determine the set of prime numbers modulo which it can be reduced. Moreover, we describe an algorithm to find an annihilating polynomial of the reduction of a hypergeometric function modulo p.
Paper Structure (19 sections, 12 theorems, 40 equations, 2 figures)

This paper contains 19 sections, 12 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our contibutions.
  3. Valuations (the $p$-adic picture)
  4. Zigzag functions
  5. Drifted valuations
  6. Recurrence over the tropical semiring
  7. Halting criterion
  8. An example
  9. Complexity
  10. Newton polygons
  11. Recurrence over $\mathcal{N}$
  12. An example
  13. Application to $p$-adic evaluation
  14. Reduction modulo primes
  15. Good reduction primes
  16. ...and 4 more sections

Key Result

lemma 1

For $k \geq 0$, we have $\text{\rm val}_p(h_k) = \sum_{r=1}^\infty w_r(k)$.

Figures (2)

  • Figure 1: The sets $J_{r,i}$ for $\boldsymbol{\alpha} = (\frac{1}{3}, \frac{4}{3})$, $\boldsymbol{\beta} = (\frac{2}{3}, 1)$ and $p = 7$
  • Figure 2: The matrices $T_r$ for the hypergeometric series $\mathcal{H}\left((\frac{1}{3}, \frac{4}{3}), (\frac{2}{3}); x\right)$ and $p = 11$

Theorems & Definitions (15)

  • lemma 1
  • proposition 1
  • lemma 2
  • lemma 3
  • lemma 4
  • proposition 2
  • lemma 5
  • lemma 6: Christol
  • definition 1
  • definition 2
  • ...and 5 more