判定二元有理函数可和性的一个算法
An Algorithm for Deciding the Summability of Bivariate Rational Functions
摘要：
设$Δ_x f(x,y)=f(x+1,y)-f(x,y)$和$Δ_y f(x,y)=f(x,y+1)-f(x,y)$是关于$x$和$y$的差分算子。一个有理函数$f(x,y)$称为可求和的，如果存在有理函数$g(x,y)$和$h(x,y)$使得$f(x,y)=Δ_x g(x,y) + Δ_y h(x,y)$。最近，Chen和Singer提出了一种判定有理函数是否可和的方法。为了在算法的意义上实现他们的方法，我们需要解决两个问题。第一个问题是确定两个二元多项式的移位等价性。我们解决了这个问题，提出了一个算法，用于计算任何两个二元多项式的色散集。第二类是求解代数闭域上的一元差分方程。通过考虑$f(x,y)$的分母在一般域上的不可约因式分解，给出了一个新的判别准则，该准则只需要求出二元差分方程的有理解.这一目标可以通过推导出一个通用的分母的合理的解决方案和程度上的分子。结合这两种算法，我们可以判定二元有理函数的可和性。
英文摘要：
Let $Δ_x f(x,y)=f(x+1,y)-f(x,y)$ and $Δ_y f(x,y)=f(x,y+1)-f(x,y)$ 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 $f(x,y)=Δ_x g(x,y) + Δ_y h(x,y)$. 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.
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