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Symbolic Summation of Multivariate Rational Functions

Shaoshi Chen, Lixin Du, Hanqian Fang

TL;DR

This work develops a comprehensive framework for symbolic summation of multivariate rational functions by marrying shift-equivalence testing with isotropy-group theory. It introduces orbital decompositions and two admissible covers to efficiently reduce summability and telescoper existence to simpler, often univariate, problems, enabling constructive certificates and operators. The key contributions include a complete solution to the discrete Picard analogue for multivariate rational functions, a systematic reduction for the existence of telescopers, and practical Maple implementations that outperform prior methods on sparse cases. The results significantly enhance the applicability of creative telescoping to multivariate settings and provide concrete, algorithmic tools for testing summability and constructing telescopers in higher dimensions.

Abstract

Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop theories, algorithms and software for the symbolic summation of multivariate functions. This paper will give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of Sato's isotropy groups of polynomials, which enables us to reduce the problems to testing the shift equivalence of polynomials. Our results provide a complete solution to the discrete analogue of Picard's problem on differential forms and can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.

Symbolic Summation of Multivariate Rational Functions

TL;DR

This work develops a comprehensive framework for symbolic summation of multivariate rational functions by marrying shift-equivalence testing with isotropy-group theory. It introduces orbital decompositions and two admissible covers to efficiently reduce summability and telescoper existence to simpler, often univariate, problems, enabling constructive certificates and operators. The key contributions include a complete solution to the discrete Picard analogue for multivariate rational functions, a systematic reduction for the existence of telescopers, and practical Maple implementations that outperform prior methods on sparse cases. The results significantly enhance the applicability of creative telescoping to multivariate settings and provide concrete, algorithmic tools for testing summability and constructing telescopers in higher dimensions.

Abstract

Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop theories, algorithms and software for the symbolic summation of multivariate functions. This paper will give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of Sato's isotropy groups of polynomials, which enables us to reduce the problems to testing the shift equivalence of polynomials. Our results provide a complete solution to the discrete analogue of Picard's problem on differential forms and can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.
Paper Structure (19 sections, 34 theorems, 140 equations, 3 figures, 3 algorithms)

This paper contains 19 sections, 34 theorems, 140 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Telescopers and summability of rational functions
  4. Complexity estimates
  5. Shift equivalence testing of polynomials
  6. Overview of the general algorithm
  7. Proof of correctness of Theorem \ref{['THM:general_cover']}
  8. Two special admissible covers
  9. Isotropy groups and orbital decompositions
  10. Isotropy groups
  11. Orbital decompositions
  12. The rational summability problem
  13. Orbital reduction for summability
  14. Summability criteria
  15. The existence problem of telescopers
  16. ...and 4 more sections

Key Result

Lemma 3.1

(See Dvir2014complete) Let $p, q \in \mathbb F[ {\bf x}]$. Then

Figures (3)

  • Figure 3.1: Terms of the polynomial $p(x+a,y+b)$
  • Figure 3.2: Graph for Lemma \ref{['LEM:DOS_sub_special_solution']}
  • Figure 3.3: Graph for Lemma \ref{['LEM:sub_special_solution']}

Theorems & Definitions (86)

  • Definition 2.1: Telescoper
  • Definition 2.3: Summability
  • Remark 2.5
  • Lemma 3.1
  • Definition 3.3: Linearization
  • Example 3.4
  • Definition 3.5: Admissible cover
  • Theorem 3.7
  • Theorem 3.8
  • proof
  • ...and 76 more