  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.
