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Computing discrete residues of rational functions

Carlos E. Arreche, Hari P. Sitaula

TL;DR

The paper develops a factorization-free algorithm to compute discrete residues of rational functions, providing an obstruction-based tool for rational summability and its extensions. By combining iterated Hermite reduction with a simple reduction that enforces squarefree, shift-free denominators, and by employing Trager-style residue computation, it yields symbolic residue data as pairs of polynomials (Bk, Dk) without explicit factorization. This enables efficient handling of multi-function and creative telescoping problems, and connects residue data to difference Galois theory, including applications to diagonal systems. The approach offers a conceptually transparent, gcd-based method that facilitates extensions to related difference frameworks and practical computation of summability-related obstructions and Galois-structural relations.

Abstract

In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational summability. More explicitly, for a given rational function f(x), there exists a rational function g(x) such that f(x) = g(x+1) - g(x) if and only if every discrete residue of f(x) is zero. Discrete residues have many important further applications beyond summability: to creative telescoping problems, thence to the determination of (differential-)algebraic relations among hypergeometric sequences, and subsequently to the computation of (differential) Galois groups of difference equations. However, the discrete residues of a rational function are defined in terms of its complete partial fraction decomposition, which makes their direct computation impractical due to the high complexity of completely factoring arbitrary denominator polynomials into linear factors. We develop a factorization-free algorithm to compute discrete residues of rational functions, relying only on gcd computations and linear algebra.

Computing discrete residues of rational functions

TL;DR

The paper develops a factorization-free algorithm to compute discrete residues of rational functions, providing an obstruction-based tool for rational summability and its extensions. By combining iterated Hermite reduction with a simple reduction that enforces squarefree, shift-free denominators, and by employing Trager-style residue computation, it yields symbolic residue data as pairs of polynomials (Bk, Dk) without explicit factorization. This enables efficient handling of multi-function and creative telescoping problems, and connects residue data to difference Galois theory, including applications to diagonal systems. The approach offers a conceptually transparent, gcd-based method that facilitates extensions to related difference frameworks and practical computation of summability-related obstructions and Galois-structural relations.

Abstract

In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational summability. More explicitly, for a given rational function f(x), there exists a rational function g(x) such that f(x) = g(x+1) - g(x) if and only if every discrete residue of f(x) is zero. Discrete residues have many important further applications beyond summability: to creative telescoping problems, thence to the determination of (differential-)algebraic relations among hypergeometric sequences, and subsequently to the computation of (differential) Galois groups of difference equations. However, the discrete residues of a rational function are defined in terms of its complete partial fraction decomposition, which makes their direct computation impractical due to the high complexity of completely factoring arbitrary denominator polynomials into linear factors. We develop a factorization-free algorithm to compute discrete residues of rational functions, relying only on gcd computations and linear algebra.
Paper Structure (12 sections, 10 theorems, 50 equations, 6 algorithms)

This paper contains 12 sections, 10 theorems, 50 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation and conventions
  4. Partial fraction decompositions
  5. Summability and dispersion
  6. Discrete residues and summability
  7. Iterated Hermite reduction
  8. Simple Reduction
  9. Computation of discrete residues
  10. Extensions and applications
  11. Connections with Galois theory
  12. Example

Key Result

Proposition 3.2

$f\in\mathbb{K}(x)$ is rationally summable if and only if $\mathrm{dres}(f,\omega,k)=0$ for every $\omega\in \overline\mathbb K$ and $k\in\mathbb N$.

Theorems & Definitions (27)

  • Definition 3.1: chen-singer:2012
  • Proposition 3.2: chen-singer:2012
  • Proposition 3.3: Abramov:1971
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Definition 5.1
  • Lemma 5.2
  • proof
  • ...and 17 more