Symbolic Summation of Multivariate Rational Functions

TL;DR

This paper develops complete algorithms for symbolic summation of multivariate rational functions, addressing the rational summability and telescoper existence problems using polynomial shift equivalence, advancing multivariate symbolic computation.

Contribution

It introduces novel solutions to two key problems in multivariate rational function summation, expanding the scope of symbolic summation algorithms beyond univariate cases.

Findings

Provides algorithms for rational summability of multivariate functions.

Offers criteria for the existence of telescopers in multivariate rational functions.

Connects the problems to polynomial shift equivalence testing.

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.