An Algorithm for Deciding the Summability of Bivariate Rational Functions

TL;DR

This paper introduces algorithms to determine whether a bivariate rational function can be expressed as a sum of differences, extending previous methods by solving polynomial shift equivalence and difference equations.

Contribution

It presents a new algorithm for deciding summability of bivariate rational functions by computing dispersion sets and solving univariate difference equations.

Findings

Algorithm for computing dispersion sets of bivariate polynomials.

New criterion for summability based on rational solutions of difference equations.

Effective method to decide summability of bivariate rational functions.

Abstract

Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$ such that $f(x,y)=\Delta_x g(x,y) + \Delta_y h(x,y)$. Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method in the sense of algorithms, we need to solve two problems. The first is to determine the shift equivalence of two bivariate polynomials. We solve this problem by presenting an algorithm for computing the dispersion sets of any two bivariate polynomials. The second is to solve a univariate difference equation in an algebraically closed field. By considering the irreducible factorization of the denominator of $f(x,y)$ in a general field, we present a new criterion which requires only finding a rational solution of a bivariate difference equation. This goal can be achieved by deriving a universal denominator of the rational solutions and a degree bound on the numerator. Combining these two algorithms, we can decide the summability of a bivariate rational function.